T  J 


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LIBRARY 


UNIVERSITY  OF  CALIFORNIA, 


RECEIVED    BY   EXCHANGE 


Class 


THE  GAS  TURBINE 


AN  " INTERNAL  COMBUSTION"  PRIME-MOVER 


BY 


SANFORD  A.  MOSS,  M.S.  (UN1V.CAL.) 


A  THESIS  PRESENTED  TO  THE  FACULTY  OF  CORNELL  UNIVERSITY 
FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


ITHACA,  N,  Y. 
MAY,  1903 


THE  GAS  TURBINE 


AN  " INTERNAL  COMBUSTION"  PRIME-MOVER 


BY 


SANFORD  A.  MOSS,  M.S.  (UNIV.  CAL.) 


A  THESIS  PRESENTED  TO  THE  FACULTY  OF  CORNELL  UNIVERSITY 
FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


.     MAY,  1903 
ANDRUS  &  CHURCH 
ITHACA,  N.  Y. 


CONTENTS. 

PAGE 

CHAPTER  i. — General  Account  of  the  Gas  Turbine 5 

CHAPTER  2. — History  of  the  Gas  Turbine 13 

CHAPTER  3. — Thermodynamics  of  the  Brayton  Cycle 17 

CHAPTER  4. — Theory  of  Nozzles  and  Free  Expansion 25 

CHAPTER  5. — Theory  of  a  Rapidly  Rotating  Disc  on  a  Flexible  Shaft__  36 

CHAPTER  6. — Experiments  with  a  Gas  Turbine. 48 


THE  GAS  TURBINE. 

AN  "INTERNAL  COMBUSTION"  PRIME  MOVER. 


CHAPTER  I. 

GENERAI,  ACCOUNT  OF  THE  GAS  TURBINE.* 

The  "  Gas  Turbine  "  is  a  form  of  heat  engine,  or  mechanism 
for  changing  the  potential  energy  of.  fuel  into  mechanical  work. 
As  we  shall  have  occasion  to  see  in  Chap.  3,  the  gas  turbine 
seems  to  offer  greater  theoretical  possibilities  than  the  steam  en- 
gine and  boiler,  or  the  Otto  cycle  gas  engine,  which  are  the  heat 
engines  now  most  usual.  Of  course  considerable  experimental 
perfecting  would  be  necessary  before  the  gas  turbine  could  de- 
mand serious  attention  for  commercial  uses.  However,  the  theo- 
retical possibilities  seem  to  warrant  the  present  investigation  at 
least. 

Before  discussing  the  gas  turbine  itself  we  shall  consider  an- 
other form  of  heat  engine  based  on  the  same  thermodynamic 
principles,  but  somewhat  simpler  in  conception. 

Consider  a  compressed  air  transmission  plant,  comprising  an 
air  compressor  and  a  motor  like  an  ordinary  steam  engine.  The 
work  done  by  the  motor  varies  directly  with  the  absolute  temper- 
ature of  the  air  supplied  to  it.  That  is  to  say,  the  work  done  by 
the  air  depends  only  on  its  volume  when  supplied  to  the  motor, 
and  hence  increasing  this  volume  by  raising  the  temperature  in- 
creases the  work,  regardless  of  the  actual  quantity  or  weight  of 
the  air.  For  this  reason  compressed  air  is  usually  "  reheated  " 
before  use  in  a  motor. 

If  the  air  is  compressed  without  any  gain  or  loss  of  heat,  that 
is,  "  adiabatically,"  the  temperature  will  rise  considerably.  In 
our  theoretical  discussion  we  will  assume  such  compression. 
Practically  it  is  always  convenient  to  cool  the  air  somewhat  dur- 
ing compression.  If  the  air  supplied  to  the  motor  is  at  the 
adiabatic  temperature  at  which  it  theoretically  leaves  the  com- 

*  The  substance  of  this  chapter  was  published  as  an  advance  portion  of 
this  thesis  in  The  Engineer  (Cleveland),  Vol.  40,  No.  8,  April  15,  1903. 


—  6  — 

pressor,  the  power  from  the  motor  will  be  theoretically  the  same 
as  that  required  for  the  compressor.  By  reheating  the  air  suffi- 
ciently, the  power  from  the  motor  may  be  made  to  exceed  the 
power  for  the  compressor  by  as  much  as  we  please.  The  excess 
power  represents  work  obtained  from  heat  energy  of  the  fuel  used 
in  reheating. 

In  a  transmission  plant,  compressor  and  motor  are  placed  at  a 
distance  from  each  other.  If,  however,  they  were  at  the  same 
place,  the  motor  could  drive  the  compressor,  and  the  excess 
power  would  be  available  for  outside  purposes.  The  apparatus 
would  then  be  a  heat  engine  operating  on  the  fuel  used  in  heat- 
ing the  air  between  compressor  and  motor. 

The  air  might  be  heated  by  passing  it  through  a  closed  vessel 
of  some  sort  exposed  to  an  outside  flame,  or  by  "  internal  com- 
bustion "  which  we  will  discuss  later.  The  entire  connected 
system  of  piping  and  vessels  between  the  compressor  and  motor, 
including  the  heating  vessel,  is  everywhere  at  the  same  pressure, 
which,  of  course,  is  the  maximum  compression  pressure.  The 
air  is  therefore  heated  at  constant  pressure,  and  its  volume  in- 
creases so  that  air  leaves  the  heating  chamber  much  faster  than 
it  enters.  The  motor  must  of  course  use  up  the  entire  volume  of 
heated  air  as  fast  as  it  is  supplied  in  order  that  the  pressure  may 
remain  constant. 

The  series  of  operations  performed  on  a  portion  of  working 
substance  as  it  goes  through  a  heat  engine  is  called  its  "cycle," 
the  cycle  used  in  the  heat  engine  just  described  being  named  the 
"  Bray  ton  cycle."  This  cycle  is  distinguished  by  the  fact  that 
while  the  heat  used  is  being  added  to  the  air,  or  other  working 
substance  which  actuates  the  engine,  this  working  substance  is 
kept  at  constant  pressure.  The  same  cycle  or  series  of  operations 
maybe  carried  out  by  very  different  kinds  of  engines.  The  gas 
turbine  operates  differently  from  the  engine  just  described,  but  it 
uses  the  same  Brayton  cycle.  That  is,  the  same  series  of  opera- 
tions are  carried  out,  but  in  a  different  way. 

Theoretically  the  Brayton  cycle  should  be  easier  to  execute 
than  the  Otto  cycle  used  in  modern  gas  engines.  The  thermo- 
dynamic  efficiency,  or  theoretical  fraction  of  the  heat  supply 
which  is  given  out  as  mechanical  work,  is  the  same  for  the  Otto 
as  for  the  Brayton  cycle,  if  the  same  compression  pressure  is 
used.  However,  the  maximum  pressure  is  very  much  greater 
with  the  Otto  cycle,  where  the  heat  is  added  at  constant  volume 
with  consequent  increase  of  pressure,  than  with  the  Brayton  cycle, 


where  the  pressure  does  not  rise  above  the  compression  pressure. 
The  maximum  temperature  is  also  much  greater  for  the  Otto 
cycle  than  for  the  Brayton,  for  the  same  compression  pressure 
and  theoretical  efficiency.  Having  lower  maximum  pressure  and 
temperature  we  should  expect  the  Brayton  cycle  to  be  mofe 
easily  managed  than  the  Otto  for  equal  efficiency. 

However,  the  method  in  which  a  cycle  is  carried  out  is  as  im- 
portant as  the  form  of  cycle.  Losses  not  taken  account  of  by 
theory  always  occur,  and  the  magnitude  of  these  depends  only  on 
the  mechanism  of  the  heat  engine  which  executes  the  cycle.  We 
must  carry  out  the  Brayton  cycle  in  an  actual  engine  without  in- 
curring greater  losses  than  occur  in  an  Otto  gas  engine  in  order 
to  secure  better  results.  In  all  actual  attempts  to  execute  the 
Brayton  cycle  in  the  past  the  losses  have  been  so  serious  as  to 
more  than  outweigh  the  theoretical  advantages.  On  paper,  at 
least,  the  gas  turbine  avoids  these  losses,  as  we  shall  see  in 
Chap.  3. 

Let  us  next  consider  how  the  Brayton  cycle  is  carried  out  by 
the  gas  turbine,  which,  as  has  been  stated,  is  the  equivalent  of 
the  compressor- motor  combination  already  discussed.  In  the 
case  of  the  compressor-motor  combination  the  compressed  air  was 
supposed  to  be  heated  by  being  passed  through  a  vessel  exposed 
to  an  outside  flame.  In  the  gas  turbine  there  is  substituted  for 
this  internal  combustion,  used  so  successfully  in  the  modern  gas 
engine.  Since  air  is  our  original  working  substance  and  since  it 
is  also  requisite  for  the  combustion  of  the  fuel  used  in  the  heating, 
we  may  use  the  air  which  is  to  operate  the  motor  to  support  the 
combustion  of  the  fuel.  That  is  to  say,  we  will  heat  the  air  for 
the  motor  by  burning  the  fuel  right  in  it.  This  is  called  "  inter- 
nal combustion,"  as  distinguished  from  "  external  combustion," 
where  the  heat  of  the  fuel  is  transferred  through  the  walls  of  a 
vessel  to  the  working  substance,  as  in  the  case  of  a  steam  boiler. 
The  air  must  be  heated  between  the  compressor  and  motor,  when 
it  is  under  considerable  pressure.  A  closed  vessel  or  enlargement 
of  the  pipe  between  the  compressor  and  motor  is  provided,  called 
the  "  combustion  chamber,"  in  which  the  air  and  fuel  combine 
or  burn,  always,  of  course,  while  at  the  maximum  compression 
pressure.  From  this  chamber  the  products  of  combustion  pass  to 
the  motor.  Theoretically  this  could  be  an  engine  like  a  steam 
engine,  or  any  substitute  for  it,  and  in  this  particular  case  we  will 
use  a  turbine  wheel  similar  to  a  steam  turbine.  The  details  of 
this  we  will  discuss  later,  first  considering  more  fully  the  method 
of  operating  the  combustion  chamber. 


—  8  — 

Heat  engines  using  the  Bray  ton  cycle  and  having  a  combustion 
chamber  with  internal  combustion,  but  using  a  motor  similar  to 
an  ordinary  steam  engine,  instead  of  a  turbine  wheel  as  in  our 
case,  have  been  built  or  planned  in  a  multitude  of  forms,  and  the 
type  of  combustion  chamber  proposed  for  any  one  of  these  is  of 
course  suitable  for  the  gas  turbine.  The  relation  of  these  engines 
to  the  gas  turbine  we  shall  discuss  more  fully  in  Chap.  2. 

In  the  Cayley  engine,  actually  in  operation  in  1807,  coal  was 
used  as  fuel.  A  supply  sufficient  for  some  time  was  placed  on  a 
grate  in  a  closed  vessel  and  lighted,  and-  the  air  from  the  com- 
pressor forced  through  it.  When  the  fuel  had  all  burned  out,  the 
motor  had  to  be  shut  down,  the  combustion  chamber  opened,  and 
a  fresh  fuel  supply  introduced.  Of  course  a  method  could  be 
easily  contrived  for  introducing  coal  into  the  combustion  chamber 
while  the  pressure  is  on,  so  that  the  motor  could  be  operated  con- 
tinuously. 

In  other  cases  a  gaseous  fuel  has  been  forced  into  the  combus- 
tion chamber  against  pressure  within  by  means  of  a  small  gas 
compressor,  and  the  air  forced  in  by  a  separate  air  compressor, 
the  two  pipes  meeting  as  they  enter  the  combustion  chamber. 
After  once  being  lighted  a  continuous  jet  of  flame  issues  from  the 
meeting  point  of  the  pipes.  The  first  gas  turbine  ever  proposed, 
which  we  will  discuss  later,  was  to  operate  in  this  way. 

The  most  convenient  fuel,  however,  for  a  gas  turbine  will  be  a 
liquid  of  some  kind,  as,  for  instance,  gasoline,  kerosene,  or  dis- 
tilled or  crude  petroleum.  Many  proposed  Brayton  cycle  engines 
with  motors  similar  to  steam  engines  have  been  planned  for 
liquid  fuel.  The  original  engine  of  this  type  was  that  of  George 
Brayton,  actually  in  commercial  use  about  1872.  It  is  due  to 
this  engine  that  the  name  of  Brayton  is  given  to  the  cycle  in 
which  heat  is  added  at  constant  pressure.  Such  a  cycle  had  been 
proposed  long  before,  however,  and  Brayton  only  invented  a  par- 
ticular form  of  engine  for  executing  the  cycle. 

The  cylinder  of  Bray  ton's  motor  was  really  his  combustion 
chamber.  He  forced  the  oil  and  air  through  passages  at  the  en- 
trance to  the  cylinder  as  the  piston  was  advancing  on  the  forward 
stroke,  and  by  means  of  a  small  flame,  kept  constantly  lighted, 
ignited  and  burned  the  mixture  as  it  entered  the  cylinder.  The 
hot  products  of  combustion,  of  course,  at  the  maximum  compres- 
sion pressure,  forced  the  motor  piston  ahead.  In  this  case  the 
combustion  was  intermittent,  the  supply  of  air  and  oil  being  cut 
off  when  the  motor  piston  reached  a  certain  point,  and  the  com- 
bustion resumed  at  the  beginning  of  the  next  stoke. 


—  9  — 

In  the  case  of  the  gas  turbine  air  and  oil  are  continuously 
forced  into  one  end  of  the  combustion  chamber,  and,  once  being 
lighted,  the  jet  continues  to  burn.  The  pressure  within  the  com- 
bustion chamber  and  the  air  pipe  leading  to  it  is,  of  course, 
always  the  maximum  compression  pressure. 

"Combustion  at  constant  pressure"  occurs  in  all  of  the  cases 
we  have  considered,  whether  the  fuel  be  solid,  liquid,  or  gaseous. 
The  fuel  being  present,  and  air  supplied  as  fast  as  necessary  by 
the  compressor,  combustion  ensues  exactly  as  if  the  fuel  were 
burned  in  the  open  air.  The  fact  that  the  combustion  takes  place 
in  a  closed  vessel  under  pressure  makes  no  difference  whatever. 
The  products  of  the  combustion  are  nitrogen  and  carbon  dioxide, 
together  with  oxygen  from  any  air  in  excess  of  that  required  for 
combustion.  All  of  the  heat  of  combustion  of  the  fuel  is  emitted 
just  as  when  it  is  burned  in  the  open  air,  and  this  goes  to  heat 
the  products  of  combustion,  which  are  thereby  expanded  so  that 
they  occupy  a  considerably  increased  volume.  Now  the  products 
of  combustion  have  practically  the  same  volume  at  any  given 
temperature  and  pressure  as  would  the  original  air.  Therefore 
the  volume  and  temperature  of  the  products  after  combustion  will 
be  the  same  as  the  original  air  would  have  if  the  entire  heat  of 
combustion  of  the  fuel  were  added  to  the  air  by  passing  it  through 
a  heated  vessel,  without  changing  the  chemical  composition  by 
burning  the  fuel  in  it.  In  effect,  then,  internal  combustion  is 
merely  a  means  of  adding  a  certain  amount  of  heat  to  the  air, 
with  consequent  increase  of  volume,  and  is  equivalent  to  passing 
it  through  a  heated  pipe.  The  change  of  chemical  composition 
is  merely  incidental.  Of  course  we  always  assume  that  the  pro- 
ducts of  combustion  are  drawn  off  from  the  combustion  chamber 
as  fast  as  they  are  formed,  so  that  the  pressure  within  it  is  always 
the  maximum  compression  pressure.  Owing  to  the  expansion 
due  to  heating,  the  volume  taken  from  the  combustion  chamber 
and  supplied  to  the  motor  is  much  greater  than  the  volume  of  air 
supplied  by  the  compressor.  As  already  stated,  the  power  ob- 
tained from  the  motor  is  greater  than  that  required  to  drive  the 
compressor  in  the  same  ratio. 

Next  let  us  consider  briefly  the  theory  of  the  turbine  wheel  or 
11  impulse  wheel  "  as  it  should  properly  be  called.  Suppose  we 
have  a  vessel  from  which  we  may  obtain  a  continuous  supply  of 
liquid  or  gas  under  pressure.  This  might  be  a  reservoir  of  water, 
a  steam  boiler,  or  the  combustion  chamber  of  a  gas  turbine.  By 
using  an  engine  similar  to  a  steam  engine  we  could  obtain  power 


IO 

from  the  pressure  of  the  liquid  or  gas.  Suppose,  however,  that 
we  connect  an  open  nozzle  with  the  vessel,  .so  that  the  water, 
steam,  or  gas  can  escape  into  the  air  or  into  any  region  where  the 
pressure  is  lower.  A  jet  will  then  issue  from  the  nozzle  with  con- 
siderable velocity.  If  the  nozzle  be  properly  shaped  so  as  to 
avoid  friction  losses,  the  kinetic  energy  of  the  escaping  jet  will  be 
exactly  the  same  as  the  energy  which  could  be  obtained  by  ex- 
panding the  same  liquid  or  gas  from  the  higher  pressure  to  the 
lower  in  a  pressure  engine  like  a  steam  engine.  That  is  to  say, 
the  velocity  with  which  the  jet  escapes  represents  exactly  the 
same  power  as  could  be  obtained  by  using  the  liquid  or  gas  in  an 
ordinary  engine,  We  will  give  the  proof  of  this,  and  consider 
the  whole  matter  in  greater  detail  in  Chap.  4. 

By  directing  the  jet  upon  the  vanes  or  buckets  of  a  properly  ar- 
ranged impulse  wheel,  the  power  of  the  jet  may  be  taken  from 
it  and  applied  to  do  useful  work.  Examples  of  this  are  shown 
by  the  Pelton  type  of  water  wheel,  and  the  De  Laval  steam  tur- 
bine. 

Suppose,  then,  that  we  place  at  the  farther  end  of  the  combus- 
tion chamber  a  proper  nozzle  and  direct  the  jet  of  gases  issuing 
from  it  upon  a  turbine  wheel.  The  wheel  could  be  of  the  type 
used  in  the  De  Laval  steam  turbine  with  a  single  set  of  buckets, 
or  a  number  of  wheels  could  be  placed  in  series,  as  with  the 
Curtis  or  Parsons  steam  turbines.  We  will  then  obtain  the  same 
power  from  the  wheel  as  we  would  if  we  used  a  motor  with  a 
piston  like  that  of  Brayton,  Cayley,  and  others.  As  already 
stated,  this  power  exceeds  the  power  required  to  drive  the  com- 
pressor by  as  much  as  the  heating  expanded  the  products  of  com- 
bustion. Hence,  we  may  drive  the  compressor  from  the  impulse 
wheel,  and  have  surplus  power,  which  is  the  net  income  from  the 
heat  energy  of  the  fuel  used. 

Having  examined  the  individual  steps  in  the  operation  of  the 
gas  turbine,  we  are  prepared  to  consider  the  apparatus  as  a 
whole.  Fig.  i  is  a  diagrammatic  sketch  of  the  complete  machine, 
which  is  so  labelled  as  to  be  sufficiently  explicit.  We  will  discuss 
possible  modifications  of  this  fundamental  arrangement  in  Chap. 
3  and  actual  details  of  construction  and  operation  in  Chap.  6. 

Oil  and  air  are  forced  into  the  combustion  chamber  as  indi- 
cated, and  begin  to  burn  at  the  entrance.  The  flaming  stream 
passes  along  the  combustion  chamber,  and  combustion  is  com- 
plete by  the  time  the  nozzle  is  reached.  The  same  constant  pres- 
sure is,  as  previously  stated,  maintained  throughout  the  system 


II 


from  the  compressor,  through  the  pipes  and  combustion  chamber 
to  the  nozzle,  since  the  nozzle  is  of  such  size  as  to  allow  the  in- 
creased volume  of  products  of  combustion  to  pass  out  of  the  com- 
bustion chamber  as  fast  as  they  are  produced  by  the  entrance  of 


FIG.  i. — Diagrammatic  Plan  of  Fundamental  Gas  Turbine. 

air  and  fuel.  The  diagram  shows  the  arrangement  for  liquid 
fluid,  to  which  the  gas  turbine  is  perhaps  best  adapted.  Gaseous 
fuel,  such  as  any  kind  of  coal  gas  or  producer  gas,  could  be  used 
by  replacing  the  oil  pump  by  a  gas  compressor.  Solid  fuel  could 
be  used  by  devising  some  means  of  continuously  feeding  it 
against  the  pressure  of  the  combustion  chamber,  or  by  having  a 
combustion  chamber  large  enough  to  contain  a  supply  of  fuel  for 
some  time,  and  forcing  in  air  only. 

The  combustion  chamber  must,  of  course,  be  designed  to  with- 
stand the  high  temperature  of  the  burning  gases  and  the  pressure 
of  the  compressor.  The  nozzle  must  be  so  designed  that  the 
products  of  combustion  have  free  expansion  from  the  maximum 
pressure  to  the  external  pressure,  hence,  as  in  the  case  of  the 
steam  turbine,  the  channel  must  first  converge  and  then  diverge. 
The  proper  shape  for  the  nozzle  we  will  consider  in  detail  in 


—  12  — 

Chap.  3.  The  gases  will  be  cooled  considerably  in  expanding 
through  the  nozzle,  but  are  still  hot  at  the  exit.  In  all  heat 
engines  only  a  portion  of  the  heat  of  the  fuel  is  utilized  and  the 
rest  must  be  thrown  away.  In  the  case  of  the  gas  turbine,  the 
kinetic  energy  of  the  jet  issuing  from  the  nozzle  represents  that 
part  of  the  heat  energy  of  the  fuel  which  can  be  utilized,  while 
the  temperature  of  the  jet  represents  that  portion  of  the  energy 
which  will  be  lost. 

The  gases  issue  from  the  nozzle  with  a  velocity  comparable 
with  that  in  the  De  Laval  steam  turbine,  and  the  turbine  must 
therefore  rotate  at  the  same  extremely  high  speed  if  we  use  a 
single  stage  wheel  as  does  De  Laval,  the  velocity  of  the  buckets 
being  approximately  half  that  of  the  jet.  The  theory  of  a  disc 
rotating  at  such  extraordinary  speeds  is  given  in  Chap.  5.  Some 
form  of  speed-reducing  mechanism  must  also  be  introduced,  since 
it  is  difficult  to  utilize  power  from  the  wheel  at  the  original  speed. 
However  the  speed  of  the  wheel  could  be  reduced  by  using 
several  wheels  in  series,  as  in  the  Curtis  or  Parsons  turbines. 
The  use  of  a  true  turbine,  where  expansion  occurs  within  the 
wheel  itself,  is  also  conceivable.  Everything  considered  how- 
ever, a  single  stage  impulse  wheel  .seems  best. 


CHAPTER  II. 

HISTORY  OF  THE  GAS  TURBINE. 

As  is  the  case  with  many  other  supposed  novelties,  the  gas  tur- 
bine was  conceived  in  ancient  times.  What  is  probably  the  orig- 
inal gas  turbine  was  patented  in  England  in  1791  by  one  John 
Barber.  The  drawing  of  this,  Fig.  2,  and  the  following  descrip- 


FIG.  2. — Barber's  Gas  Turbine,  1791. 


—  i4  — 

tion  are  taken  from  the  British  Patent  Reports,  Volume  XXIII, 
1791-2,  Patent  No.  1833,  which  seems  to  be  the  only  source  of 
information  concerning  the  curious  machine.  It  will  be  seen  that 
Barber  probably  understood  thoroughly  the  principle  of  the  gas 
turbine,  although  the  mechanical  knowledge  of  the  time  would 
hardly  have  sufficed  for  the  construction  of  a  working  machine. 

The  vessels  marked  i  are  retorts  for  the  production  of  the  gas 
to  be  used,  by  distillation  of  coal,  wood,  etc.,  by  means  of  an  ex- 
ternal flame.  These  are  in  duplicate,  so  that  one  can  always  be  in 
use  while  the  other  is  being  emptied  of  coke  and  recharged.  The 
vessel  above  the  retorts  is  a  cooling  and  condensing  chamber, 
from  which  the  gas  is  drawn  by  the  pipe  B.  The  apparatus  up 
to  this  point  is  merely  for  the  production  of  the  gaseous  fuel  to  be 
used,  and  is  not  an  essential  feature  of  the  gas  turbine.  The 
parts  marked  CC,  DD,  the  two  lower  vessels  of  those  marked  4, 
and  the  elevated  water  tanks  above  the  apparatus  comprise  two 
peculiar  hydraulic  compressors,  the  details  of  which  we  need  not 
investigate.  The  triangular  shaped  vessel  between  the -tanks  4 
and  the  wheels  is  the  combustion  chamber.  By  means  of  the 
front  compressor,  the  gas  taken  from  B  is  compressed  and  dis- 
charged from  the  front  vessel  4  into  the  combustion  chamber  by 
the  pipe  shown.  By  means  of  the  rear  compressor  air  is  com- 
pressed and  discharged  from  the  rear  vessel  4  by  a  pipe  not  shown 
into  the  rear  side  of  the  combustion  chamber.  The  upper  vessel 
4  discharges  water  into  the  combustion  chamber,  which  absorbs 
some  of  the  heat  of  combustion  and  reduces  the  maximum  tem- 
perature. This  is  not  essential  to  the  operation  of  the  apparatus, 
however,  as  the  same  result  could,  of  course,  be  obtained  by  the 
use  of  an  excess  of  air. 

The  air  and  gas  must  burn  within  the  combustion  chamber, 
and  the  products,  issuing  from  the  lower  end,  are  directed  upon 
vanes  of  the  wheel  marked  8,  which  is  thereby  impelled  to  rotate. 
Barber's  description  of  the  action  within  the  combustion  chamber 
is  not  very  explicit,  and  there  is  a  possibility  that  he  did  not  un- 
derstand the  gas  turbine  after  all.  It  may  be  that  the  triangular 
chamber  was  only  intended  as  a  mixing  chamber,  and  that  the 
combustion  was  not  to  occur  until  the  mixture  had  passed  through 
its  outlet,  since  the  flame  would  not  strike  back  owing  to  the  ve- 
•locity  through  the  orifice.  It  might  be,  on  the  other  hand,  that 
the  air  and  gas  were  to  be  throttled  before  they  entered  the  cham- 
ber, so  that  atmospheric  pressure  existed  within  it.  In  either 
event  the  heat  would  be  added  while  the  working  substance  was 


at  atmospheric  pressure,  instead  of  while  it  was  at  the  maximum 
pressure.  The  addition  of  heat  would  then  be  of  no  use  what- 
ever, and  the  wheel  8  would  operate  equally  well  whether  the 
gases  were  lighted  or  not.  Theoretically  the  power  required  for 
compression  would  then  be  equal  to  the  power  of  the  turbine 
wheel  ;  and  practically,  owing  to  friction,  etc.,  would  be  in  excess  ; 
so  that  the  machine  would  not  operate  at  all. 

In  case,  however,  that  Barber  had  the  correct  conception  of  the 
matter  and  arranged  to  have  the  combustion  take  place  in  the  tri- 
angular chamber,  where  also  the  maximum  compression  pressure 
existed,  then  the  operation  would  be,  as  he  states,  as  follows  :  A 
pinion  on  the  impulse  wheel  shaft  operates  the  gear  wheel  10,  on 
the  shaft  of  which  are  the  cams  9,  which  raise  rods  attached  to 
the  walking  beams  55,  actuating  the  compressors  by  means  of  the 
chains  shown.  A  pinion  on  the  same  shaft  as  the  wheel  10  ope- 
rates the  upper  gear  wheel.  From  the  projecting  shaft  of  this  up- 
per gear  the  difference  between  the  compressor  and  motor  power 
can  be  taken  off  and  applied  to  useful  work. 

The  maximum  compression  pressure  which  Barber  could  obtain 
was  that  due  to  the  head  of  water  from  the  tanks  in  the  upper 
part  of  the  figure,  which  practically  could  not  be  made  very  great. 
A  modern  gas  turbine  would  use  much  greater  pressures  than  Bar- 
ber contemplated. 

Nothing  seems  to  have  come  of  Barber's  gas  turbine  and  the 
matter  lay  untouched  for  years.  In  the  meantime  a  great  number 
of  internal  combustion  Brayton  cycle  engines  with  reciprocating 
engines  as  motors,  were  proposed  as  already  stated,  none  attain- 
ing success  however.  A  number  of  these  are  described  by  Mr. 
Charles  Lucke  in  a  paper  presented  to  the  American  Society  of 
Mechanical  Engineers,  Dec.  190^,  andjorming  part  of  his  Doc- 
tor's thesis,  presented  to  Columbia  University  in  1902.  He  dis- 
tinguishes two  types,  the  first  type  having  intermittent,  and  the 
second  type  continuous  internal  combustion  at  constant  pressure. 
The  latter  type  of  course  involves  a  combustion  chamber  or  re- 
ceiver sufficiently  large  to  prevent  excessive  fluctuations  of 
pressure  owing  to  the  intermittent  demand  of  the  motor.  A  heat 
engine  of  this  type  is  essentially  equivalent  to  the  gas  turbine 
with  the  single  exception  that  a  reciprocating  motor  is  used 
instead  of  an  impulse  wheel.  Mr.  Lucke  mentions  the  possibility 
and  even  the  desirability  of  using  a  "gas-expansion  turbine" 
instead  of  a  reciprocating  motor  in  this  type  of  heat  engine,  but 
he  does  not  go  into  details. 


—  i6  — 

A  number  of  gas  turbines  have  been  proposed  by  various  in- 
ventors in  which  an  explosive  mixture  is  repeatedly  admitted  to 
a  chamber,  exploded,  and  discharged  with  gradually  decreasing 
pressure  upon  the  vanes  of  a  turbine  wheel.  These  might  be 
called  "  intermittent  combustion"  gas  turbines.  A  gas  turbine 
of  this  type  due  to  Leon  Le  Pontois  was  the  subject  of  the  Doc- 
tor's thesis  presented  to  Cornell  University  by  Mr.  W.  O. 
Amsler.  All  gas  turbines  of  this  character  seem  to  be  based 
on  the  false  idea  that  "explosion"  is  the  only  proper  way 
to  conduct  internal  combustion,  and  -the  inventors  seem  to 
be  ignorant  of  the  possibility  of  continuous  combustion  under 
constant  pressure.  For  obvious  reasons  a  continuous  combustion 
gas  turbine  is  immeasurably  superior  to  one  with  intermittent 
combustion  and  variable  pressure. 

The  writer  began  the  study  of  the  gas  turbine  in  1898,  believ- 
ing the  idea  original  Until  research  disclosed  Barber's  patent. 
A  thesis  for  the  degree  M.S.  entitled  "Thermodynamics  of  the 
Gas  Turbine"  was  presented  to  the  University  of  California  in 
1900,  the  substance  of  which  is  given  in  the  next  chapter. 


CHAPTER  III. 

THERMODYNAMICS  OF  THE   BRAYTON  CYCI,E. 

The  heat  engine  used  in  the  Gas  Turbine,  commonly  called  the 
Brayton  Cycle,  as  already  stated,  is  distinguished  by  the  fact  that 
heat  is  added  to  and  taken  from  the  working  substance  while  it  is 
at  constant  pressure.  The  various  operations  performed  on  a 
portion  of  working  substance  as  it  passes  through  a  Brayton  cycle 
heat  engine  are  as  follows  : 

The  working  substance,  initially  separate  portions  of  air  and 
fuel  in  the  case  of  the  gas  turbine,  is  first  compressed  to  the  maxi- 
mum pressure  pv  While  at  this  pressure  heat  is  added  by  some 
means  or  other  ;  in  the  case  of  the  gas  turbine  by  "internal  com- 
bustion. ' '  The  volume  and  temperature  of  the  working  substance 
are  thereby  increased.  The  chemical  composition  may  also  be 
changed  but  this  is  merely  on  incident  which  we  have  on  occasion 
to  take  account  of  at  present.  The  working  substance  next  ex- 
pands adiabatically  to  atmospheric  pressure.  During  addition  of 
heat  and  expansion,  work  is  done  which  is  eventually  utilized  by 
the  motor.  In  our  case  an  impulse  wheel  is  used,  which  is,  how- 
ever, as  will  be  shown  in  Chap.  4,  equivalent  to  a  piston  engine. 
The  working  substance,  now  at  atmospheric  pressure  but  at  a 
considerable  temperature,  is  next  discarded,  and  a  new  lot  taken 
into  the  compressor  at  atmospheric  temperature  and  pressure.  In 
order  to  close  the  thermodynamic  cycle  we  may  suppose  that  the 
same  working  substance  is  used  over  again,  enough  heat  having 
been  extracted  at  constant  pressure  to  reduce  the  temperature 
appropriately. 

The  mathematical  treatment  of  an  actual  cycle  such  as  the 
above  has  never  been  attempted  directly,  and  the  best  that  can  be 
done  is  to  discuss  the  matter  on  the  assumption  that  the  working 
substance  is  a  perfect  gas  throughout.  The  approximations  that 
this  assumption  involves  in  the  case  of  the  gas  turbine  are  as  fol- 
lows :  The  working  substance  is  initially  separate  portions  of  oil 
and  air.  Since  the  work  of  bringing  the  oil  to  the  maximum 
pressure  is  insignificant,  the  actual  work  of  compression  is  not 
quite  so  great  as  we  take  it  to  be  by  assuming  the  working  sub- 
stance wholly  gaseous.  However,  as  we  shall  see  later,  in  cases 
where  the  proportion  of  oil  is  reduced  sufficiently  to  give  work- 
able temperatures,  the  oil  forms  two  or  three  per  cent,  by  weight  of 


—  i8  — 

the  working  substance.  Hence  but  slight  error  is  committed  by 
assuming  that  the  work  of  pumping  the  oil  is  equal  to  the  work  of 
compressing  on  equal  weight  of  air. 

In  the  next  place  the  variation  of  volume  due  to  addition  of 
heat  by  internal  combustion  is  very  irregular,  and  does  not  follow 
the  simple  law  of  the  perfect  gas  cycle,  where  the  volume  varies 
directly  with  the  heat  added.  Some  irregularities,  probably 
slight  owing  to  the  small  percentage  of  oil,  arise  from  the  fact 
that  some  of  the  heat  of  combustion  of  the  oil  is  absorbed  by  its 
latent  heat  of  vaporization.  This  heat  causes  increase  of  volume 
but  not  of  temperature.  Other  irregularities  arise  from  the  "ex- 
pansion "  due  to  combustion.  When  the  volume  of  a  perfectly 
combustible  mixture  of  gases  is  compared  with  volume  of  the 
products  of  combustion  ;  both  calculated  by  Avogadro's  law,  and 
reduced  to  the  same  pressure  and  temperature,  an  expansion  or 
contraction  is  found  to  have  occurred.  It  can  be  shown  that  ex- 
pansion always  occurs  when  a  hydrocarbon  Ca  Hm  burns,  if  m  is 
greater  than  ^.  Therefore,  since  most  of  the  hydrocarbons  of 
which  oil  is  composed  are  quite  complex,  an  expansion  occurs 
due  to  the  change  of  chemical  composition.  This  is  independent 
of  the  heating  of  the  products  of  combustion  due  to  the  addition 
of  the  heat  of  combustion.  This  is  therefore  another  source  of 
increase  of  volume  without  change  of  temperature.  Usually 
there  is  a  considerable  amount  of  dilution  in  the  mixtures  used, 
particularly  on  account  of  the  nitrogen  of  the  air,  so  that  the  ex- 
pansion is  not  a  very  large  percentage  of  the  whole  volume  in- 
volved, making  this  irregularity  slight  also. 

The  increase  of  volume  due  to  the  two  causes  referred  to  of 
course  represents  work  done  due  to  liberation  of  the  chemical 
energy  of  the  fuel  It  may  be  remarked  that  the  work  thus  done 
is  probably  never  included  as  it  ought  to  be  in  the  tabulated 
values  of  the  heat  of  combustion  of  fuels.  The  "  constant  press- 
ure calorimeter  "  of  the  Junkers'  or  other  form,  which  is  usually 
used,  takes  no  account  of  energy  liberated  by  combustion  but 
causing  volume  variation  merely  without  incidental  change  of 
temperature. 

The  differences  between  the  actual  cycle  and  the  assumed  per- 
fect gas  cycle  have  thus  far  been  found  slight.  There  is,  how- 
ever, another  difference  of  much  greater  magnitude.  Recent  re- 
search has  probably  established  the  fact  that  the  specific  heat  of 
all  gases  increases  with  the  temperature.  Therefore  the  specific 
heat  of  the  products  of  combustion  at  the  high  temperatures 


attained  in  the  gas  turbine  is  much  greater  than  the  constant 
value  assumed  in  the  perfect  gas  cycle.  It  follows  that  the  actual 
temperatures  attained  in  the  gas  turbine  are  much  less  than  those 
calculated  on  the  perfect  gas  assumption.  Since  so  much  approxi- 
mation is  involved  in  the  perfect  gas  assumption,  we  may  as  well 
assume  in  addition  that  the  working  substance  has  throughout 
the  specific  heat  of  air  at  atmospheric  temperatures. 

Let  us  consider  the  effect  of  the  "imperfect"  nature  of  the 
actual  gases  upon  the  theoretical  efficiency  of  the  cycle.  As  is 
well  known,  in  the  case  of  the  Carnot  cycle  imperfection  of  the 
working  fluid  has  no  effect  UROII  the  theoretical  efficiency.  It 
may  also  be  shown  that  the  theoretical  efficiency  is  independent 
of  the  nature  of  the  working  fluid  for  any  cycle  in  which  heat  is 
added  and  taken  away  along  lines  of  the  same  constant  specific 
heat*  It  does  not  appear  therefore  as  if  the  imperfection  of  the 
working  substance  would  have  any  marked  effect  on  the  theo- 
retical efficiency  in  the  case  of  our  cycle.  That  is  to  say  we  will 
make  the  hypothesis  that  the  theoretical  efficiency  of  the  actual 
gas  turbine  is  approximately  the  same  as  that  calculated  on  the 
basis  that  the  working  substance  is  a  perfect  gas. 

The  algebraical  and  arithmetical  calculations  for  a  number  of 
cases  of  perfect  gas  Brayton  cycles  comprised  the  master's  thesis 
of  the  writer,  entitled  "  Thermodynamics  of  the  Gas  Turbine," 
filed  at  the  University  of  California.  A  brief  abstract  will  be 
given  of  the  most  important  results. 

The  heat  Q  added  by  the  combustion,  per  pound  of  working 
substance,  may  be  as  a  maximum  the  heat  emitted  by  the  com- 
bustion of  as  much  oil  as  is  necessary  to  form  a  perfectly  com- 
bustible mixture  of  air  and  oil  amounting  to  one  pound.  Ordi- 
nary crude  or  refined  petroleum  is  about  85  per  cent,  carbon  and 
15  per  cent.  Hydrogen.  Calculation  from  this  analysis  gives 
about  15  pounds  of  air  as  necessary  to  burn  i  pound  of  oil,  with 
emission  of  about  20,000  B.T  U.  Hence  a  perfect  mixture  of 

air    and   oil   weighing    i   pound    emits  2O'OC^  —   1250   B.   T.   U. 

However  in  order  to  be  sure  that  no  oil  will  be  unconsumed  there 
should  be  a  slight  excess  of  air,  so  that  we  will  assume  a  round 
figure  of  i  ,000  as  the  greatest  desirable  value  of  Q.  By  decreas- 
ing the  proportion  of  oil  to  air  we  may  make  Q  anything  that  we 
please  under  1,000.  Of  course  this  figure  may  need  modification 
if  fuels  other  than  oil  are  used. 

*See  Generalization  of  Carnot' s  Cycle,  Physical  Review,  Vol.  16,  No.  i, 
Jan.,  1903. 


—  20 — 

The  temperature  and  volume  after  combustion  are  calculated 
from  the  value  of  Q  and  the  specific  heat  at  constant  pressure  for 
air,  '238,  assumed  as  constant  as  already  explained.  The  final 
temperature  and  volume  are  calculated  by  assuming  adiabatic 
expansion  from  maximum  to  atmospheric  pressure.  The  heat  q 
which  must  be  extracted  to  return  the  working  substance  to  the 
initial  condition  is  found  from  the  specific  heat  at  constant 
volume,  as  before.  The  heat  transformed  into  mechanical  work 
is  the  difference  Q  —  q  so  that  the  efficiency  of  the  cycle  which 

we  will  call  e  is  given  by  e  =  — 2. 

The  work  done  in  compression  is  C  and  the  work  done  by  the 
motor  is  M,  so  that  the  net  work  available  for  useful  purposes  is 
C  —  M.  Of  course  C—  M  =  J  (Q —  q}  where /is  the  mechani- 
cal equivalent  of  heat.  A  measure  of  the  complication  involved 
in  the  necessary  compression  of  the  air  before  combustion  will  be 
given  by  the  ratio  of  the  power  required  for  compression,  to  the 
net  power,  which  we  will  call  r. 

Then  r=      £--. 

C — M 

In  deciding  the  values  of  the  various  quantities  to  be  used  in  a 
particular  Brayton  Cycle,  we  may  choose  arbitrarily  the  maxi- 
mum pressure  pl  and  the  heat  added  per  pound  of  working  sub- 
stance Q.  The  latter,  however,  must  never  exceed  1000  as  already 
stated.  When/j  and  Q  are  decided,  everything  else  may  be  cal- 
culated directly.  On  the  other  hand,  we  may  assign  arbitrary 
values  to  any  other  two  quantities  whatever,  and  calculate  the 
values  of  pl  and  Q  which  must  necessarily  be  used  to  .secure  them. 

In  cases  where  the  cycle  of  a  heat  engine  is  executed  in  a  motor 
involving  a  reciprocating  piston,  the  maximum  pressure  and  tem- 
perature must  be  brought  within  reasonable  limits,  so  that  these 
are  made  the  arbitrarily  determined  variables.  In  the  case  where 
ah  "impulse  wheel"  is  used  as  a  motor  for  the  Brayton  Cycle, 
giving  a  Gas  Turbine  ;  the  temperature  at  the  end  of  expansion 
and  the  linear  velocity  of  the  impulse  wheel  must  be  within  con- 
trollable limits.  The  peripheral  velocity  of  a  De  Laval  Steam 
Turbine,  a  Pelton  water  wheel,  or  any  similar  impulse  wheel, 
should  theoretically  be  one-half,  and  practically  about  three- 
eighths  of  the  velocity  of  the  jet  which  drives  it.  The  velocity  of 
the  jet  Fis  such  that  the  kinetic  energy  of  a  pound  of  working 

F2 
substance,  — ,  is,  as  already  stated,   equal  to  the  work   which 


—  21  — 

would  be  done  by  the  same  pound  in  expanding  behind  a  piston, 
given  by  M.  Hence  the  assignment  of  a  value  to  the  velocity  of 
the  impulse  wheel  of  a  gas  turbine  amounts  to  assigning  a  value 
to  M.  An  extreme  value  for  the  velocity  is  100,000  feet  per 
minute. 

We  shall  next  consider  some  numerical  results  for  a  number  of 
cases  of  the  Bray  ton  Cycle,  given  in  Table  i.  We  will  among 
other  things  give  the  speed  which  the  impulse  wheel  should  have 
in  case  the  cycle  is  executed  by  using  a  gas  turbine.  Of  course 
all  of  the  other  values  hold  good  for  any  form  of  engine  whatever 
in  which  the  cycle  is  executed,  whether  a  reciprocating  piston 
engine  or  whatnot.  As  previously  stated,  we  will  assume  the 
working  substance  as  a  perfect  gas,  so  the  temperatures  we  obtain 
are  very  much  higher  than  will  occur  in  an  actual  case. 

Case  I  is  a  theoretical  Brayton  Cycle  in  which  compression  and 
expansion  are  both  adiabatic.  We  will  suppose  a  perfect  ma- 
chine with  no  losses. 

Case  II  is  a  modification  of  the  Brayton  Cycle,  in  which  the 
compression  is  isothermal  and  a  regenerator  is  used.  We  will 
again  assume  no  losses.  We  will  find  that  the  regenerator  gives 
a  remarkable  gain.  The  formulas  show  that  the  work  done  by 
the  motor  is  equal  to  the  heat  added  by  combustion.  Hence  we 
obtain  as  useful  work  all  of  the  heat  of  combustion  except  the 
work  required  for  compression.  The  less  the  compression  pres- 
sure the  less  this  lost  work  and  therefore  the  greater  the  efficiency. 
This  is  not  the  case,  however,  if  any  losses  occur.  The  theo- 
retical efficiencies  for  Case  II  are  remarkably  high,  but  are  of 
course  unattainable  since  we  have  assumed  a  perfect  heat  engine. 

Case  III  is  a  recalculation  of  Case  II  in  which  we  attempt  to 
find  values  for  the  efficiency  which  might  be  expected  in  an 
actual  case  by  assuming  the  various  losses  which  will  occur. 
We  will  assume  that  the  motor  only  gives  70%  of  the  power 
which  it  theoretically  should,  that  the  compressor  requires  20% 
more  power  than  it  should,  and  that  the  regenerator  has  an  effi- 
ciency of  60%.  It  appears  that  we  can  in  no  case  use  the  full 
value  looo  for  the  heat  added  per  pound  of  working  substance, 
since  the  temperatures  will  be  too  high  for  a  reciprocating  engine 
and  the  velocities  too  high  for  a  Gas  Turbine.  An  excess  of  air 
is  therefore  used  to  reduce  the  heat  added  per  pound. 

The  values  given  for  the  efficiency  in  this  case  are  remarkably 
high,  and  since  they  are  results  which  might  be  expected  in  an 
actual  engine,  they  are  worthy  of  attention.  Of  course  experi- 


—  22  — 


merit  will  be  necessary  to  determine  just  what  values  of  tempera- 
ture, etc.,  are  practicable.  As  stated  the  values  of  the  tempera- 
tures given  in  Table  i  do  not  hold  for  the  actual  cycle. 

TABLE  i. — BRAYTON  CYCI,E  CALCULATIONS. 


1 

IH 

5 

*7  bo 

(4-4 

o 

£§' 

3   cO 

i, 

i 

1 

o 

'§ 

oD 

%  *> 
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CASE  I. 

90 
90 

IOOO 

250 

O 
O 

4665 
1505 

2435 
652 

.22 

.87 

115,300 
71,680 

•43 
•43 

Adiabatic  Compression. 
Perfect  Machine  with 
No  Losses. 

195 

495 
495 

IOOO 

250 

IOOO 

250 

o 

0 
0 
0 

4860 
1710 
5190 
2040 

2008 
546 
1562 

434 

.27 
1.07 

•34 
1.38 

131,2001  .54 

83,740    .54 
147,600    .64 
98,300    .64 

CASE  II. 

90 

IOOO 

o 

9239 

5039 

;o8 

159,100    .93 

Isothermal  Compression 
with  Regenerator. 
Perfect  Machine  with 

90 

195 
195 

495 

250 

IOOO 

250 

IOOO 

o 
o 
o 
o 

1965 
7376 
1498 
6087 

915 
3176 
448 
1887 

•39 
.1  1 
.61 
•  15 

79,370 
159,100 
79,570 
159,100 

.72 

•91 
.62 

•87 

No  Losses. 

495 

250 

o 

1176 

126 

1.03 

79,570 

•49 

CASE  III. 

56 

333 

0 

2147 

1200 

•  74 

75,4oo 

.27 

Iso.  Comp.  with  Regen. 
Actual  Machine  with 
Assumed  Losses. 
Air  Excess. 

82 
61 
98 

33 
61 

392 

445 
545 
445 
569 

0 

o 
o 
o 
o 

2398 
2867 
3233 
3179 
3699 

I2OO 
1619 
1619 
2139 
2139 

.68 
•50 
•47 
.40 
•36 

84,800 
86,485 
99,960 
78,155 
96,775 

•30 

•35 
.28 

•33 

CASE  IV. 

Iso.  Comp.  with  Regen. 

79 

625 

•37 

1275 

600 

•67 

75,000 

•  13 

Assumed  Losses. 

47 

638 

.36 

1680 

IOOO 

•43 

75,ooo 

.15 

Cooling  Water. 

The  excess  of  air  required  to  keep  the  temperature  down  in  the 
above  case  demands  an  increased  size  of  compressor.  However, 
the  temperature  could  be  kept  down  by  the  use  of  water,  which 
can  be  introduced  more  readily.  Case  IV  is  therefore  one  in 
which  the  working  substance  is  a  mixture  of  air  and  water,  the 
air  being  just  sufficient  for  perfect  combustion.  The  water  be- 
comes superheated  steam  at  the  same  temperature  as  the  products 
of  combustion  and  the  mixture  operates  the  motor.  We  suppose 
isothermal  compression  of  the  necessary  air,  and  a  regenerator  as 


-23- 

before.  The  regenerator  can  only  utilize  the  sensible  heat  of  the 
steam  however,  and  must  reject  the  latent  heat.  This  lowers  the 
efficiency,  and  therefore  this  case  is  not  as  favorable  as  Case  III. 
The  compressor  is  much  smaller  in  proportion  however,  as  shown 
by  the  small  value  of  r.  The  assumed  efficiencies  of  compressor 
motor  and  regenerator  were  needlessly  taken  somewhat  differently 
in  this  case  than  in  the  preceding. 

The  efficiencies  of  Case  III  are  so  much  higher  than  those  of 
Case  IV  that  it  probably  will  be  best  to  use  air  in  excess  rather 
than  water,  in  order  to  give  controllable  temperatures. 

As  stated,  we  have  attempted  to  take  account  of  all  possible 
losses  in  calculating  Case  III  "ley  assuming  dynamical  efficiencies 
for  the  various  parts  of  the  apparatus.  It  is  to  be  noted  that  we 
do  not  have  any  extra-thermodynamic  or  thermal  losses  in  the  gas 
turbine,  which  prove  so  serious  in  heat  engines  with  reciprocating 
motors,  such  as  the  steam  engine  and  Otto  gas  engine.  The  most 
important  source  of  loss  not  taken  account  of  by  theory  in  a  gas 
engine  is  the  loss  of  heat  through  the  cylinder  walls  to  the  jacket 
water.  There  is  no  such  loss  in  the  case  of  the  gas  turbine,  as 
the  maximum  temperature  occurs  within  a  chamber  where  there 
are  no  moving  parts,  so  that  cooling  is  unnecessary.  By  lining 
the  combustion  chamber  thickly,  or  covering  it  with  a  non-con- 
ductor, the  temperature  at  the  outside,  and  consequently  the 
radiation  loss,  may  be  made  very  small,  even  though  very  high 
temperatures  exist  within. 

One  source  of  dynamic  loss  in  the  gas  turbine  is  the  friction  as 
the  gases  pass  through  the  nozzle  and  as  they  pass  along  the 
buckets  of  the  impulse  wheel.  The  steam  turbine  is  subject  to 
exactly  similar  losses,  and  the  fact  that  they  have  not  proven 
serious  in  its  development  indicates  that  they  will  probably  not 
be  serious  in  the  case  of  the  gas  turbine.  We  have  attempted  to 
take  account  of  the  loss  in  Case  III  by  assuming  an  efficiency  of 
70%  for  the  motor. 

Another  source  of  dynamic  loss  in  the  gas  turbine  is  that  occur- 
ring in  the  compressor,  since  its  "mechanical  efficiency"  is. 
always  less  than  unity.  We  have  attempted  to  take  account  of 
this  loss  in  Case  III  by  assuming  that  the  compressor  requires 
20%  more  power  than  it  theoretically  should. 

It  is  to  be  noted  that  losses  in  compressor  and  motor  which  are 
comparatively  small  percentages  of  the  powers  of  these  machines 
are  much  larger  percentages  of  the  net  power,  which  is  the  differ- 
ence between  the  two.  Suppose,  for  instance,  that  the  compres- 


—  24  — 

sor  requires  theoretically  10  and  actually  12  horsepower,  and  that 
the  motor  yields  theoretically  50  but  actually  35  horsepower. 
The  net  power  will  then  be  theoretically  40  and  actually  only  23. 
That  is  to  say  a  20%  loss  in  the  compressor  and  a  30%  loss  in  the 
motor  cause  a  loss  of  42^  %  of  the  theoretical  net  power.  It  is 
probably  due  to  this  great  effect  on  the  net  power  of  compara- 
tively small  compressor  and  motor  losses  that  the  Brayton  cycle 
has  never  been  really  successful  in  a  reciprocating  engine,  al- 
though tried  many  times.  A  reciprocating  motor  necessarily  has 
a  water  jacket,  and  this  adds  possibly  a  50%  thermal  loss  to  the 
dynamic  motor  loss.  The  net  power  is  therefore  enormously  re- 
duced. In  the  gas  turbine  we  escape  the  thermal  loss  and  thus 
have  some  possibility  of  obtaining  reasonably  high  values  of  the 
net  power.  This  is  shown  by  the  net  efficiencies  of  Case  III, 
Table  i,  which  have  been  calculated  with  reasonable  values  of 
the  various  losses  taken  into  account,  as  already  stated.  These 
efficiencies  compare  so  favorably  with  those  obtained  in  the  mod- 
ern steam  and  gas  engine  that  it  seems  quite  possible  that  after 
sufficient  experiment  and  development  the  gas  turbine  may  trans- 
form heat  into  work  more  efficientlv  than  either  of  these. 


CHAPTER  IV. 

THEORY  OF  NOZZLES  AND   FREE  EXPANSION.* 

Suppose  that  a  portion  of  fluid,  liquid  or  gaseous,  with  a  given 
volume  and  temperature,  exists  in  a  region  where  a  given  pres- 
sure is  continuously  maintained.  The  fluid  then  represents  a 
certain  amount  of  stored  or  "potential"  energy.  Part  of  this 
exists  as  heat  energy  due  to  the  temperature  of  the  portion,  and 
the  balance  is  due  to  its  presence  in  a  region  where  the  pressure 
is  continuously  maintained.  -If  the  volume  leaves  the  region, 
work  will  be  done,  the  amount  being  given  by  the  product  of 
pressure  and  volume.  Hence  the  presence  of  the  portion  of  fluid 
in  the  region  under  pressure  implies  stored  energy  of  this  amount. 

Suppose  that  in  some  way  or  other  our  portion  of  fluid  reaches 
a  region  of  lower  pressure,  the  change  being  made  without  add- 
ing or  extracting  heat,  that  is  "  adiabatically."  The  potential 
energy  will  then  be  found  to  be  less  than  before.  The  difference 
has  left  the  gas  and  must  be  sought  elsewhere. 

There  are  two  ways  in  which  the  portion  of  fluid  may  pass 
from  one  pressure  to  another.  In  the  first  place,  it  may  be 
placed  behind  a  piston  and  allowed  to  expand  gradually,  as  when 
steam  expands  in  a  steam  engine.  Then  the  difference  between 
the  potential  energies  for  the  two  pressures  appears  as  the  me- 
chanical work  done  by  the  piston.  This  may  be  called  "con- 
strained expansion. ' '  In  the  second  place,  the  portion  of  fluid  may 
expand  from  one  pressure  to  the  other  by  passing  through  a 
"  nozzle  "  or  orifice  in  the  wall  of  the  vessel  in  which  the  higher 
pressure  exists,  into  the  region  where  the  lower  pressure  exists. 
This  is  called  "free  expansion."  The  result  of  free  expansion  is 
the  reduction  of  the  portion  of  fluid  to  the  same  final  condition  as 
regards  pressure,  volume  and  temperature  as  in  the  case  of  con- 
strained expansion,  so  that  the  loss  of  potential  energy  must  be 
the  same. 

Now  each  small  portion  of  the  fluid,  while  in  the  orifice,  is 
always  passing  from  one  point  to  another  where  the  pressure  is 
slightly  lower.  At  any  instant,  therefore,  any  small  portion  is 
acted  on  by  a  force  due  to  the  difference  between  the  pressures 
behind  it  and  before  it.  This  force  does  work  in  increasing  the 
velocity  of  the  portion  of  fluid  considered,  so  that  the  mass  ac- 

*The  substance  of  this  chapter  was  published  as  an  advance  portion  of 
this  thesis  in  Sibley  Journal,  Vol.  17,  No.  6,  March,  1903. 


—  26  — 

quires  a  gradually  increasing  stock  of  kinetic  energy  as  it  moves 
through  the  orifice.  The  kinetic  energy  of  the  fluid  as  it  finally 
leaves  the  orifice,  together  with  any  energy  dissipated  in  the 
form  of  heat  due  to  friction  against  the  sides  of  the  orifice,  repre- 
sents the  difference  in  the  potential  energy  of  the  fluid  when  at 
the  higher  and  when  at  the  lower  pressure.  That  is  to  say,  in 
free  expansion  we  have  kinetic  energy  corresponding  to  the  work 
done  on  the  piston  in  constrained  expansion. 

In  both  cases,  however,  work  is  done  representing  the  differ- 
ence between  the  potential  energies  corresponding  to  the  two 
pressures.  The  kinetic  energy  due  to  free  expansion  is  therefore 
exactly  the  same  as  the  amount  of  work  which  would  be  done  on 
an  engine  piston  by  the  same  fluid  expanding  between  the  same 
pressures. 

The  kinetic  energy  of  a  jet  of  fluid  issuing  from  an  orifice  after 
free  expansion  may  be  taken  from  the  fluid  and  applied  to  do  use- 
ful work  by  means  of  an  "  impulse  wheel  "  having  a  number  of 
vanes  or  buckets  on  which  the  jet  impinges.  Any  form  of  motor 
in  which  a  piston  is  moved  by  a  fluid  under  pressure  may  there- 
fore have  substituted  for  it  an  impulse  wheel  driven  by  a  jet  of 
the  fluid  issuing  from  a  vessel  under  the  same  initial  pressure.  A 
hydraulic  pressure  engine  may  therefore  be  replaced  by  a  Pelton 
water  wheel,  a  steam  engine  by  a  De  Laval  steam  turbine,  and  a 
Brayton  Gas  Engine  by  a  "  gas  turbine." 

The  kinetic  energy  of  a  jet  of  gas  after  free  expansion  is  usually 
enormous  compared  with  the  mass,  so  that  very  high  velocities 
are  attained.  The  friction  losses  may  therefore  become  serious  if 
the  nozzle  is  not  properly  designed.  On  the  other  hand  the 
friction  losses  are  comparatively  small  with  proper  precautions. 
A  requisite  for  minimizing  friction  loss  is  that  the  velocity  in  the 
nozzle  shall  gradually  and  continuously  increase,  since  sudden 
variations  cause  shock.  As  the  increase  of  velocity  is  caused  by 
the  decrease  of  pressure  from  point  to  point,  the  pressure  should 
decrease  gradually  and  continuously. 

In  the  case  of  water,  the  volume  does  not  vary  with  the 
pressure,  and  hence  the  velocity  at  any  point  along  a  water  nozzle 
depends  only  on  the  cross-section  at  that  point,  regardless  of  the 
pressure.  Hence  a  continuous  increase  of  velocity  is  secured  by  a 
continuous  decrease  of  section.  That  is  to  say  a  nozzle  for  water 
should  converge  continuously  from  the  point  of  greatest  pressure 
to  the  point  of  least  pressure. 

A  nozzle  for  a  gas  must  have  a  radically  different  shape  in  order 
that  the  velocity  may  increase  and  the  pressure  decrease  con- 


tinuously.  The  volume  of  a  given  mass  of  gas  increases  as  pres- 
sure decreases  and  this  increase  of  volume  must  be  taken  into  ac- 
count in  conjunction  with  the  desired  increase  of  velocity  in  order 
to  find  the  nozzle  shape.  Now  the  nozzle  must  converge  in  order 
to  give  an  increase  of  velocity,  and  diverge  in  order  to  allow  for  a 
change  of  volume  due  to  decrease  of  pressure.  If  the  velocity  is 
to  increase  faster  than  the  volume  increases,  the  nozzle  will  con- 
verge on  the  whole.  However,  it  is  quite  possible  that  the  pressure 
may  decrease  so  that  the  consequent  rate  of  increase  of  volume  is 
greater  than  the  consequent  rate  of  increase  of  velocity.  Then 
the  nozzle  must  diverge  in  order  to  provide  for  the  greater  rate  at 
which  the  volume  increases.  \ 

As  we  will  show  presently  a  decrease  of  pressure  down  to  about 
one-half  of  the  initial  pressure  causes  a  greater  rate  of  increase  of 
velocity  than  of  volume,  requiring  the  nozzle  to  converge.  If  the 
pressure  is  to  decrease  still  further  the  volume  begins  to  increase 
more  rapidly  than  the  velocity  and  hence  the  nozzle  must  diverge. 
That  is  to  say  when  the  final  pressure  is  less  than  about  one-half 
of  the  initial  pressure  the  nozzle  first  converges  and  then  diverges 
in  order  to  allow  for  a  continuous  decrease  of  pressure. 

If  the  nozzle  is  not  shaped  in  this  manner  the  pressure  cannot 
decrease  continuously  down  to  the  final  pressure.  That  is  to  say 
a  convergent  nozzle  is  an  abnormal  shape  for  a  gas,  and  gives  a 
very  abnormal  action  which  we  will  later  briefly  investigate.  A 
nozzle  first  convergent  and  then  divergent  is  the  normal  shape  for 
a  continuous  decrease  of  pressure  and  nothing  abnormal  or 
mysterious  occurs  when  such  a  nozzle  is  used.  It  is  often  popu- 
larly supposed  that  the  free  expansion  of  gases  is  very  mysterious 
owing  to  the  irregular  action  in  a  convergent  nozzle,  whereas  this 
irregularity  is  due  solely  to  the  fact  that  a  convergent  nozzle  is 
proper  only  for  non-expansible  fluids  such  as  water. 

With  this  introduction  we  may  proceed  to  treat  our  subject 
mathematically.  We  will  take  the  case  of  a  perfect  gas  although, 
as  already  stated,  the  gases  used  in  a  gas  turbine  are  by  no  means 
perfect.  However  the  results  will  apply  approximately.  Our 
results  also  hold  approximately  for  the  case  of  superheated  steam, 
now  commonly  used  in  steam  turbines,  and  for  compressed  air 
issuing  from  an  orifice.  The  same  methods  would  be  used  for  the 
case  of  saturated  steam,  but  the  equations  would  be  somewhat 
different.  We  will  neglect  friction  altogether.  The  gas  passes 
through  a  nozzle  from  a  region  where  the  pressure  is  p^  pounds 
per  square  foot  to  a  region  where  the  pressure  is  zero.  Since  the 


—  28  — 

pressure  is  to  decrease  continuous!}'  from  />,  to  O  we  may  cut  the 
nozzle  off  at  a  point  where  the  pressure  has  diminished  to/3  and 
thus  obtain  a  nozzle  suited  for  discharge  into  a  region  of  pres- 
sure />3.  Let  the  general  conditions  at  any  point  whatever  along 
the  nozzle  be  denoted  by  letters  without  subscripts,  p  being  the 
absolute  pressure  in  pounds  per  square  foot,  v  the  corresponding 
specific  volume  in  cubic  feet  per  pound,  T  the  absolute  Fahrenheit 
temperature,  Fthe  velocity  in  feet  per  second,  and  A  the  nozzle 
area  in  square  feet. 

Let  the  conditions  at  the  entrance  of  the  nozzle  be  denoted  by 
subscriptj,  at  the  point  of  minimum  diameter  by  subscript  2,  at 
the  point  where  any  particular  pressure  (as  for  instance  atmo- 
spheric pressure)  is  reached,  by  subscript  8,  and  at  the  point 
where  zero  pressure  would  be  reached  if  the  nozzle  continued  so 
far,  by  subscript  4  .  Let  M  be  the  weight  of  gas  passing  through 
in  pounds  per  second.  Let  <:p  and  cv  be  the  specific  heats  of  the 
gas  and  k  their  ratio.  R  is  the  gas  equation  constant,  equal  to 

^,    and  /  is  the  mechanical  equivalent  of  heat.     As  is  well 

known  R  =J(  c^  —  cv  ). 

Let  us  consider  that  portion  of  the  nozzle  between  the  entrance 
and  the  general  point  where  the  pressure  is  p.  The  mathematical 
statement  of  the  fact  that  no  energy  is  created  within  this  space, 
(neglecting  the  kinetic  energy  of  the  gas  entering  the  nozzle)  is 


The  neglect  of  the  entering  kinetic  energy  amounts  to  the  as- 
sumption that  the  entrance  velocity  is  zero,  so  that  the  nozzle 
area  at  the  entrance  must  be  infinite.  Practically,  however,  the 
entrance  area  may  be  comparatively  small  and  the  entrance 
velocity  be  in  the  neighborhood  of  100  to  200  feet  per  second,  and 
yet  the  kinetic  energy  of  the  entering  gas  will  be  quite  negligible 
as  compared  with  the  other  terms  of  the  expression. 
The  above  expression  reduces  to 

~r=P.vl-pv+Jcv(Tl-T) 

Since  plvl=  R  T»     pv=RT,      and 
this  becomes 


Hence  we  have  as  a  general  expression  for  the  velocity 


—  29  — 

V-VigJc^Tt-T)  (0 

We  will  assume,  as  is  usual  in  similar  cases,  that  the  gas  ex 
pands  adiabatically  in  passing  through  the  nozzle. 
Hence  the  usual  adiabatic  relation  holds, 


T, 
Therefore 


This  expression  shows  the  relation  between  the  pressure  and 
velocity  at  any  point.  It  will  be  seen  that  the  velocity  always  in- 
creases as  the  pressure  decreases. 

We  may  express  the  relation  between  the  nozzle  area  and  the 
pressure  at  any  point  by  substituting  for  Fin  (2).  From  the  re- 
lation A  V  =  Mv  and  the  adiabatic  relation 


,   we  have  V- 
v' 


Then 


r. 

This  gives  us  a  means  of  investigating  the  variation  of  the  area 
A  as  the  pressure/  varies.     Differentiating  with  respect  to/, 


where  B  is  a  group  of  coefficients  all  of  which  are  essentially  posi- 
tive. Suppose  that  /  is  equal  to /j.  Then  since  k  is  about  1.41, 

— —  will  be  positive.    will  continue  positive,  that  is  the  noz- 

dp  dp 

zle  area  will  decrease  as  the  pressure  decreases,  while  the  pressure 
varies  from/!  to  a  value  such  that  the  parenthesis  on  the  right 
vanishes.  That  is  to  say  the  nozzle  must  converge  while  the 
pressure  varies  from  pl  to  a  value  /2  such  that 

'-'  (5) 


Evidently  — -  vanishes  for/  — /2  and  hence  A  has  a  minimum 
dp 

value  for  this  pressure.     As  the  pressure  /  still  further  decreases 

—  is  a  negative  and  hence  the  nozzle  diverges. 
dp 


—  30  — 

That  is  to  say  a  nozzle  in  which  the  pressure  continuously  de- 
creases has  a  point  of  minimum  area  at  the  place  where  the  pres- 
sure is/>2  and  diverges  beyond  this  point.  If  the  shape  is  differ- 
ent from  this,  the  pressure  cannot  decrease  continuously  and  an 
irregular  action  occurs. 

In  the  case  of  air  and  a  number  of  other  gases  k=  1.41  and 
the  pressure  at  the  throat  or  point  of  minimum  area  is  /2  = 

•527^1- 
By  substituting  the  value  of  the  throat  pressure  in  the  adiabatic 

relation  between  pressure  and  temperature,  the  throat  temperature 
is  found  to  be 

T   —       2        T 
T+l      l 

By  substituting  this  value  for  T2  in  the  general  expression  for  the 
velocity  (i)  we  have  as  the  throat  velocity 


This  may  be  shown  to  be  the  velocity  of  sound  in  the  gas  for  the 
conditions  of  pressure  and  temperature  obtaining  at  the  throat. 
It  is  therefore  the  velocity  at  which  a  disturbance  will  be  pro- 
pagated along  the  stream  flowing  through  the  nozzle.  Since 
the  velocity  of  motion  beyond  the  throat  is  greater  than  the 
velocity  at  which  a  disturbance  can  be  propagated,  it  is  impossible 
for  a  disturbance  to  be  propagated  backward  along  the  stream  of 
gas.  Hence  if  the  portion  of  the  nozzle  beyond  the  throat  were 
to  be  removed,  so  that  the  gas  was  discharged  at  a  pressure  p^ 
into  a  region  of  lower  pressure  />3  ,  the  disturbance  thus  produced 
could  not  in  any  way  affect  the  flow  of  gas  between  the  entrance 
and  the  throat.  That  is  to  say  if  a  nozzle  is  wholly  convergent  and 
the  region  into  which  it  discharges  has  a  pressure  less  than  /2  the 
discharge  will  nevertheless  be  at  pressure  p2  and  the  gas  will  ex- 
pand from  /„  to  />3  by  diverging  in  the  region  beyond  the  end  of 
the  nozzle,  just  as  if  the  proper  divergent  portion  of  the  nozzle 
existed  beyond  the  throat.  This  explanation  of  the  well  known 
fact  is  in  essence  due  to  Professor  Osborne  Reynolds. 

In  order  to  compute  the  sizes  necessary  for  the  design  of  a 
nozzle  we  proceed  as  follows  :  By  substituting  the  value  of  p9 
given  by  (5)  in  the  general  expression  (4)  for  the  area  we  obtain 
a  value  for  the  throat  area  which  reduces  to 


\ 


A        \      2gk 

For  the  case  of  air  and  those  gases  for  which  k=  1.41  the  throat 
area  becomes 


In  order  to  most  conveniently  compute  the  area  and  velocity  at 
the  end  of  the  nozzle  where  it  discharges  into,  a  region  at  pressure 
/3  we  first  find  the  final  temperature  and  specific  volume  by  sub- 
stituting the  value  of  p3  in  the  general  adiabatic  relations,  whence 


We  may  then  obtain  the  final  velocity  by  substituting  the  value 
of  7~3  in  the  general  expression  for  the  velocity  (i),  which  gives 


Also,         v. 

The  final  area  may  then  be  found  by  substituting  the  values  thus 
found  for  F3  and  v3  in  the  general  expression  A  V—  Mv. 

By  taking  p  =  o  in  (4)  it  appears  that  the  final  area  for  dis- 
charge into  an  absolute  vacuum  is  infinite.  This  is  du'e  to  the 
fact  that  the  specific  volume  is  infinite  for  zero  pressure.  How- 
ever, the  area  increases  very  rapidly  for  a  very  slight  decrease  in 
pressure  in  this  vicinity,  and  the  final  area  for  what  is  practical!}' 
considered  a  vacuum  is  not  particularly  great.  It  may  be  re- 
marked that  the  final  temperature  is  zero  for  discharge  into  a 
vacuum. 

The  theory  thus  far  has  been  known  in  essence  for  some  time, 
although  the  present  method  of  treatment  is  possibly  new.  We 
have  found,  however,  only  the  entrance,  throat,  and  final  diame- 
ters of  the  nozzle.  The  curvature  of  the  nozzle  and  the  distance 
between  these  points,  is  not  determined.  The  following  attempt 
to  complete  the  discussion  is  believed  to  be  original. 

We  shall  assume  as  a  basis  for  procedure  the  manner  in  which 
a  particle  of  gas  is  to  be  accelerated  as  it  passes  through  the  noz- 
zle. Evidently  in  order  to  avoid  impact  loss  due  to  changes  of 
velocity  the  acceleration  should  proceed  in  some  regular  manner. 
One  possible  assumption  is  uniform  acceleration.  That  is  to  say 
the  force  furnished  by  the  gradually  decreasing  pressure  which 
serves  to  increase  the  velocity  may  be  taken  as  constant.  We 
will  presently  work  out  the  shape  of  a  nozzle  to  secure  this  result. 
However,  this  is  not  necessarily  the  shape  giving  a  minimum 
friction  loss.  For  instance  it  is  conceivable  that  the  losses  might 
be  less  if  the  particles  of  gas  were  accelerated  less  rapidly  as  their 


—  32- 

velocity  increased.  Then  the  accelerating  force  would  vary  in- 
versely as  the  velocity.  Theoretical  considerations  may  be  dis- 
covered which  will  give  some  absolute  criterion  as  to  the  best 
method  of  acceleration.  However,  the  matter  must  probably  be 
decided  by  experiments  with  nozzles  designed  according  to  vari- 
ous reasonable  assumptions. 

The  assumption  of  uniform  acceleration  seems  most  reasonable 
and  therefore  we  will  take  it  up  here. 

Let  x  be  the  variable  distance  from  the  nozzle  entrance  to  any 
point  where  the  pressure  is  p.  Then  x^  will  be  the  throat  dis- 
tance, xs  the  total  length  of  the  nozzle,  and  x±  the  length  for  dis- 
charge into  a  vacuum.  We  have  by  hypothesis 

<Tx  = 
dt* 

where  C  is  a  constant  to  be  determined  presently.     Also 

V=  — 
dt 

Let  us  substitute  from  these  expressions  in  the  identity 


dt  dt* 

The  substitution  gives 

—  F2  =  2C 
dx 

Integrating  and  inserting  the  obvious  value  o  for  the  integration 
constant, 

V*  =  2  Cx.  (6) 

We  evaluate  the  constant  C  by  assuming  a  value  for  x9t  the  length 
of  the  nozzle  for  discharge  at  pressure  ps.  This  must  be  decided 
by  experience,  being  long  enough  to  avoid  losses  due  to  too  rapid 
acceleration,  and  short  enough  to  avoid  excessive  friction  against 
the  nozzle  walls.  Having  computed  F3  in  the  manner  already 
described  we  have,  by  substitution  in  the  above  expression, 

c=YL 

2*3 

By  substituting  the  value  for  V.A  obtained  from  the  general  ex- 
pression (2),  we  have  as  an  equivalent  expression  for  C, 

Next,  let  us  express  the  relation  between  the  pressure  at  any 
point  and  the  distance  to  it.     By  substituting  in  (2)  the  value  of 
]/  from  (6)  we  have 


ZLEIWITh    UNIFORM  ACCELERATIO 


FIG.  3. — Nozzle  Section  with  Pressure,  Temperature  and  Velocity  Curves. 


—  34  — 
-gjf*  T,  \l_(P_ 


We  have  taken  arbitrarily  x3  the  length  of  nozzle  for  discharge 
into  a  region  of  pressure  ps.  However  the  nozzle  up  to  x^  will  be 
the  same  if  we  suppose  that  the  nozzle  is  long  enough  to  discharge 
into  a  vacuum,  and  that  the  particular  pressure  p.A  is  reached  at 
the  arbitrary  intermediate  point  x^.  The  total  length  of  the  noz- 
zle will  then  be  x^  which  is  found  by  placing  p  =  o  in  the  above 
expression.  Then 


We  shall  find  it  convenient  to  use  x±  simply  as  an  abbreviation. 
Then  the  relation  between  x  and  p  is 

*-,Jx-(A^l  (9) 


Next,  let  us  express  the  area  at  any  point  in  terms  of  the  dis- 
tance to  that  point.  By  substituting  the  value  of  Ffrom  (6)  and 
the  value  of  p  from  (9)  in  (3),  we  have 


A 


do) 


This  is  the  equation  giving  the  nozzle  shape  in  order  that  the 
gas  passing  through  may  be  uniformly  accelerated.  Curve  i,  Fig. 
3  shows  the  section  of  a  nozzle  for  a  particular  case  constructed  by 
this  equation.  The  final  pressure  is  zero,  but  by  cutting  off  the 
nozzle  at  the  proper  point  the  discharge  pressure  may  have  any 
value  that  we  please. 

It  must  be  remarked  that  the  equations  deduced  give  the  areas 
perpendicular  to  the  stream  tines,  or  lines  of  flow.  When  the 
area  is  small  compared  with  the  length,  these  are  practically  par- 
allel to  the  axis  and  the  areas  may  be  taken  on  normal  plane  sec- 
tions of  the  nozzle.  When  the  nozzle  begins  to  diverge  consider- 
ably near  the  end  the  stream  lines  spread  out  and  the  areas  given 
by  the  equation  can  no  longer  be  taken  on  plane  surfaces. 

If  we  express  A  in  full  by  substituting  the  values  of  the  con- 
stants C  and  xt  from  (7)  and  (8)  it  will  be  found  that  A  is  directly 
proportional  to  ^/and  \/Tr  inversely  proportional  to  p^  and  also 

involves  the  ratio  (  £$-  ) .      It  follows,   therefore,    that   a   nozzle 


-35- 

constructed  for  a  given  initial  pressure  and  temperature  may  have 
these  quantities  varied  and  still  be  of  the  correct  shape  for  uniform 
acceleration.  The  quantity  of  gas  passing  through,  M,  will  change 

however,  since  —  -  —  l  is   constant   for   a    given    shape:      Also, 
Pi 

PA.  1  must  be  constant  for  a  given  shape,  so  that  the  final  pres- 

P^J 

sure  will  vary  directly  with  the  initial  pressure  for  a  given  length 
of  nozzle  xa  and  a  given  shape. 

By  differentiating  (10)  and  putting  --  =o  we  obtain  the  fol- 

%  d  x 

lowing  value  for  xtt  the  distance  to  the  throat  or  point  of  minimum 
area  : 


That  is  to  say,  the  throat  is  at  a  constant  fraction  of  the  length 
of  a  complete  nozzle,  regardless  of  conditions.  This  value  of  JC2 
substituted  in  (10)  will  give  the  throat  area  A2  previously  found. 
It  will  be  interesting  to  note  the  variation  of  pressure,  tempera- 
ture and  velocity  as  we  proceed  from  point  to  point  along  the  noz- 
zle. (6)  gives  the  variation  of  velocity  from  point  to  point  and 
(9)  the  variation  of  pressure.  By  substituting  in  (9)  the  adiabatic 
relation  between  pressure  and  temperature,  we  obtain  as  the  tem- 
perature variation 


This  shows  that  the  temperature  decreases  uniformly  for  our  case 
of  uniform  acceleration. 

'From  these  equations  the  values  of  pressure,  temperature  and 
velocity  for  the  nozzle  of  Curve  i,  Fig.  3  are  plotted  in  Curves 
2,  3,  4,  each  value  being  laid  off  opposite  the  point  of  Curve  i,  to 
which  it  corresponds. 


CHAPTER  V. 

THEORY  OF  A  RAPIDLY  ROTATING  DISC  ON  A  FI<F,XIBI,E  SHAFT. 

One  of  the  possible  methods  of  operating  a  gas  turbine  is  the 
use  of  an  apparatus  similar  to  that  of  the  De  Laval  steam  tur- 
bine, involving  a  single  wheel  rotating  at  a  very  high  speed. 
The  wheel  must  of  course  be  very  carefully  balanced,  so  that  the 
center  of  gravity  coincides  as  nearly  as  possible  with  the  center 
of  the  hole  bored  to  fit  the  shaft  on  which  the  wheel  is  mounted. 
Even  with  most  careful  workmanship,  however,  the  center  of 
gravity  is  always  far  enough  away  from  the  bore  center  to  give 
rise  to  centrifugal  forces  of  considerable  magnitude  at  the  high 
rotative  speeds  necessary.  The  evil  effect  of  this  lack  of  balance 
is  obviated  by  Gustav  De  Laval's  highly  ingenious  invention  of 
a  flexible  shaft.  As  the  action  of  the  flexible  shaft  is  very  com- 
monly misunderstood,  and  as  the  full  mathematical  theory  has 
probably  never  been  given,  it  will  be  here  discussed  as  a  matter 
of  incidental  interest. 

We  will  use  the  following  notation  throughout  this  chapter. 
O  =  center  of  rotation. 
B  =  center  of  bore,  provided  the  shaft  is  initially  straight.     In 

general,  B  is  the  point  coinciding  with  O  when  there  is  no 

load  on  the  shaft. 
C  =  center  of  gravity  of  disc. 
a  =  distance  between  center  of  bore  and  center  of  gravity  =  BC. 

This  is  the  amount  by  which  the  shaft  is  out  of  balance.    It 

is  always  a  small  quantity. 
K=  force  required  to  deflect  the  shaft  unit  distance.     This  is  a 

function  of  the  dimensions  and  material  of  the  shaft  and 

character  of  the  supports.     For  ordinary  bearings  the  shaft 

is  a  beam  fixed  at  the  ends.     If  there  is  a  swivel  bearing  at 

one  or  both  ends,  the  beam  is  supported  merely. 
M=  weight  of  disc.     The  weight  of  .the  shaft  is  assumed  to  be 

neglegible. 
w  =  angular  velocity  of  disc  in  radians  per  second.     This  is  the 

angular  velocity  of  B  about  O,  and  is  supposed  to  be  main- 

tained constant  by  some  external  agency. 
*  =  period  which  the  disc  would  have  if  the  shaft  were  set  into 

transverse  vibration.    Then  it  may  be  shown  that  \l/  —    V 

\ 


. 

M 


—  37  — 

where  g  is  the  gravitation  constant.     The  time  of  a  complete 
vibration  is  then  — . 

R  =  distance  OC  between  center  of  gravity  and  center  of  rotation 
for  equilibrium.    We  will  find  later  R  = 


L  =  twisting  moment  required  to  twist  the  shaft  and  disc  through 

one  radian. 

/  =  moment  of  inertia  of  the  disc  about  its  center  of  gravity. 
<£  =  period  which  the  disc  would  have  if  the  shaft  were  set  into 

torsional  vibration.     Then  it  may  be  shown  that  <j>  =    \~JL 

The  time  of  a  complete  vibration  is  ?-rr. 

<p 

Fig.  4  represents  the  ordinary  case  of  a  shaft  slightly  out  of 
balance.  The  centrifugal  force  deflects  the  shaft  until  the  inward 
force  due  to  the  deflection  is  equal  to  the  centrifugal  force.  We 
neglect  gravity  for  the  present,  and  may  suppose  the  shaft  verti- 
cal. The  centrifugal  force  acting  on  the  center  of  gravity  is 

—  r  w2.     This  gives  rise  to  a  rotating  force  of  the  same  magnitude 
g 

pulling  on  the  bearings  and  tending  to  cause  their  vibration. 
The  shaft  deflection  is  O  B  =  r  —  a.     The  condition  that  the  force 
due  to  the  deflection  is  equal  to  the  centrifugal  force  is 

Mrv?  =  K(r—  a) 
g 
which  reduces  to 

a 

(i) 


-0' 


For  ordinary  shafts  $  is  large,  so  that  for  moderate  speeds 
is  less  than  unity.  Then  the  greater  w  the  greater  will  be  r. 
That  is  to  say,  the  greater  the  speed  the  greater  will  be  the  shaft 
deflection,  and  the  more  violent  the  rotating  pull  on  the  bearings, 
given  by 


MJ       K 

This  is  the  theory  of  the  well  known  difficulties  occurring  with 
high  speed  machinery.  The  remedy  is  to  make  M  the  weight  of 
the  rotating  parts,  as  small  as  possible,  K  which  gives  the  stiff- 


-38- 


ness  of  the  shaft,  as  great  as  possible,  and  a  which  gives  the  ec- 
centricity of  balancing,  as  small  as  possible. 

If  the  angular  velocity  be  made  greater,  the  deflection  will  in- 
crease, and  finally,  if  the  angular  velocity  w  be  made  equal  to  ^, 
the  deflection  will  theoretically  become  infinity.  This  will 
not  actually  occur,  however,  as  the  law  of  proportionality  of 
load  and  deflection  does  not  hold  except  for  small  deflections,  and 
because  vibrations  of  period  ^  which  may  occur,  will  also  have  a 


period  o>.  \j/  is  called  the  "critical  speed,"  and  considerable  dis- 
turbance occurs  however  when  o>  reaches  this  value.  No  mathe- 
matical account  has  ever  been  given  of  the  exact  action. 

Suppose  now  that  w  is  made  to  exceed  the  critical  speed.  Then 
the  denominator  of  the  expression  (i)  becomes  negative,  and  in 
order  to  have  a  positive  value  for  r  we  must  make  a  negative  also. 
This  is  done  by  taking  it  in  a  direction  opposite  that  of  Fig.  4,  so 
that  the  center  of  gravity  is  inside  of  the  center  of  bore,  as  in  Fig. 
5.  Fig.  5  therefore  represents  the  position  of  equilibrium  when  w 
is  greater  than  the  critical  speed  if/.  If  we  count  a  as  essentially 
positive  in  this  position,  the  condition  for  equilibrium  is 
MR^__ 

g 
which  reduces  to 

o_  « 

(2) 


a] 


-39- 

Here  the  greater  the  angular  velocity  to  the  less  will  be  R.  That 
is  to  say,  the  center  of  gravity  actually  comes  nearer  to  the  center 
of  rotation  as  the  speed  increases.  For  any  given  rotative  speed, 

we  secure  this  result  by  making  tf  =  — £  as   small   as   possible, 

M 

which  is  accomplished  by  making  K,  which  gives  the  stiffness  of 
the  shaft,  as  small  as  possible.  That  is  to  say,  the  more  flexible 
the  shaft  the  nearer  the  center  of  gravity  comes  to  the  center  of 
rotation.  The  center  of  gravity  always  rotates  in  a  circle  whose 
radius  is  given  by  (2),  however.  The  popular  statement  that  "  a 
flexible  shaft  allows  the  disc  to'rotate  about  its  center  of  gravity  " 
is  therefore  erroneous.  If  the  disc  were  to  rotate  about  its  .center 
of  gravity,  the  shaft  would  be  deflected  a  distance  a,  so  that  there 
would  be  an  unbalanced  deflection  force  a  K. 

The  rotating  force  pulling  on  the  bearings  and  tending  to  vi- 
brate them,  for  the  case  of  Fig.  5  is 


g  J  _     g 

K     Mrf 

This  is  diminished  and  the  machine  made  to  run  with  slight  vi- 
bration by  making  K,  which  gives  the  stiffness  of  the  shaft,  as 
small  as  possible  and  Mthe  weight  of  the  disc,  and  o>  the  angu- 
lar velocity,  as  great  as  possible.  This  is  all  expressed  by  the 
statement  that  w  -i-  \f/  should  be  as  great  as  possible.  That  is  to 
say,  the  working  value  of  the  speed  should  bear  as  great  a  ratio 
as  possible  to  the  critical  speed  in  order  to  reduce  vibration  of  the 
bearings. 

The  position  of  Fig.  4  is  sometimes  called  rotation  with  the 
"heavy  side  out"  while  that  of  Fig.  5  is  rotation  with  the 
"  heavy  side  in."  That  is  to  say,  if  a  mark  be  made  during  rota- 
tion by  touching  the  portion  of  the  edge  of  the  disc  furthest  from 
the  center  of  rotation,  in  order  to  find  the  place  to  cut  away  metal 
in  order  to  correct  the  balancing,  then  metal  must  be  taken  from 
the  side  of  disc  nearest  the  mark  if  the  speed  is  below  the  critical 
speed,  and  from  the  side  away  from  the  mark  if  the  speed  is 
above  the  critical  speed. 

The  critical  speed  in  radians  per  second  we  have  found  to  be 

t  =    \Kg.    This  reduces  to  —     [32.16  X  12  k  =  ,8?  -6     \~k 

\|     M  27T^J  M  ^-^r 

revolutions  per  minute,  where  k  is  the  load  in  pounds  required  to 
deflect  the  shaft  one  inch  and  M  is  the  total  load  in  pounds  sup- 
posed concentrated  at  the  shaft  center. 


—  4°  — 

As  we  have  seen,  Fig.  5  shows  a  position  of  equilibrium.  We 
have  next  to  show  that  the  equilibrum  is  stable.  That  is  to  say, 
we  must  show  that  if  any  accidental  cause  displaces  the  center  of 
gravity  from  the  position  shown,  it  will  return  and  not  seek  some 
new  position.*  The  balance  of  the  chapter  will  be  a  tedious 
proof  of  this  fact. 

In  discussing  the  motion  of  the  disc  we  will  make  use  of  the 
well  known  fact  that  the  center  of  gravity  C  moves  as  if  all  of 
the  forces  acting  on  the  disc  were  applied  at  it,  the  mass  being 
concentrated  there  also  ;  and  that  the  disc  rotates  about  the  center 
of  gravity  as  if  it  wrere  fixed  in  space. 

First  we  will  consider  the  motion  of  the  center  of  gravity  only, 
and  will  assume  that  the  shaft  is  vertical.  Suppose  that  the  disc 
is  not  rotating  so  that  the  B  Fig.  5  will  coincide  with  the  point  O. 
Then  the  center  of  gravity  C  will  assume  a  position  C  slightly 
below  O. 

If  the  point  B  be  displaced  in  any  direction  from  O  the  elastic- 
ity of  the  shaft  will  cause  a  force  proportional  to  the  displacement 
tending  to  return  B  to  O,  We  will  suppose  the  disc  displaced 
without  rotation,  so  that  the  center  of  gravity  C  will  have  an  ex- 
actly equal  displacement  from  its  position  of  rest  C  '  .  Since  the 
center  of  gravity  moves  as  if  the  displacement  force  acted  on  it, 
we  may  consider  that  the  center  of  gravity  C  is  acted  on  by  a  force 
proportional  to  its  displacement  from  C  '.  The  center  of  gravity 
will  then  have  a  simple  harmonic  motion,  and  will  vibrate  back 
and  forth  until  the  energy  is  absorbed  by  molecular  friction.  The 


period  of  such  a  vibration  is  well  known  to  be  $  =&  using  the 

notation  already  given. 

The  most  general  case  of  harmonic  motion  will  be  when  two 
displacements  at  different  angles  and  at  different  times  are  given 
to  the  disc.  It  will  then  vibrate  in  an  ellipse  with  period  \j/. 

We  may  now  apply  any  other  forces  to  the  shaft  when  it  has 
this  vibration,  and  the  disc  will  then  have  a  displacement  com- 
pounded of  that  due  to  the  vibration  and  that  due  to  the  new  forces. 
Suppose,  for  instance,  we  apply  a  pull  to  the  center  of  gravity  of 
constant  magnitude  but  of  varying  direction,  rotating  in  fact  with 

*  The  discussion  up  to  this  point  substantially  as  here  given,  was  published 
by  Stodola  in  Zeitschrift  des  Vereines  Deutscher  Ingenieure,  Vol.  47,  Nos. 
2  and  4,  Jan.  10  and  24,  1903.  However,  the  matter  is  given  here  since  it 
was  prepared  independently  before  the  above  publication.  The  balance  of 
this  chapter  is  probably  new.  Stodola  gives  a  treatment  along  totally  dif- 
ferent lines  which  does  not  seem  satisfactory. 


an  angular  velocity  o>.  So  far  as  the  shaft  is  concerned  this  force 
will  cause  a  rotating  displacement,  to  be  compounded  with  the 
vibrating  displacement.  Such  a  rotating  force  is  furnished  by  the 
"centrifugal  force,"  which  is  in  equilibrium  with  the  force  due 
to  a  rotating  displacement  R  +  a  of  the  shaft  center  B.  Hence 
the  disc  may  rotate  in  the  position  shown  by  Fig.  5,  and  at  the 
the  same  time  have  an  elliptical  vibration. 

Conversely,  if  the  disc  be  rotating  in  the  position  of  Fig.  5,  and 
any  accidental  displacement  be  given  to  it,  it  will  at  once  begin 
to  vibrate,  in  addition  to  the  rotation,  in  the  same  way  as  it  would 
if  the  displacement  were  given  to  it  when  at  rest. 

The  above  reasoning  is  perfectly  rigid,  but  nevertheless  the  step 
is  perhaps  a  large  one  to  make  at  once.  We  will  therefore  at  a 
later  period,  for  verification,  form  the  differential  equation  of  mo- 
tion of  the  disc,  and  show  that  the  integral  gives  a  displacement 
compounded  of  an  elliptical  vibration  and  a  rotating  displace- 
ment R.  Such  a  mathematical  treatment  hides  the  true  nature 
of  the  actions,  however,  and  the  method  above  used  is  much  more 
luminous. 

It  must  be  noted  that  if  a  displacement  from  the  position  of  Fig. 
5  be  gived  the  disc  during  rotation,  the  resulting  vibration  will 
always  be  parallel  to  the  original  displacement,  and  will  not  rotate 
in  direction.  That  is  to  say,  if  a  displacement  should  happen  to 
be  made  in  the  direction  of  the  instantaneous  position  of  the  radius 
to  the  center  of  gravity,  the  vibration  induced  will  not  continue  to 
be  radial,  since  then  the  direction  of  vibration  would  constantly 
change,  but  it  will  be  in  a  direction  fixed  in  space  parallel  to  the 
initial  direction. 

If  the  elastic  force  resulting  from  an  initially  radial  displace- 
ment were  always  radial,  it  can  be  shown  that  the  position  of  Fig. 
5  would  be  unstable. 

Any  accidental  displacement  from  the  position  of  Fig.  5  will 
therefore  in  general  cause  the  point  C  to  vibrate  in  an  ellipse  about 
the  equilibrium  position,  the  axes  of  the  ellipse  always  remaining 
parallel  to  their  original  position  in  space.  Owing  to  molecular 
friction  this  vibration  will  soon  die  away,  the  size  of  the  ellipse 
gradually  decreasing  until  the  position  of  Fig.  5  is  again  reached. 

Next  let  us  consider  the  effect  of  the  vibrations  just  considered 
in  causing  rotation  of  the  disc  about  the  center  of  gravity.  Let 
Fig.  6  represent  any  instantaneous  position  of  the  points  O,  Z?and 
C  while  C  is  making  an  elliptical  vibration.  The  path  of  B  will 
be  parallel  since  the  vibrational  displacements  have  involved  no 


—  42  — 


rotation  of  the  disc.  The  net  force  acting  on  the  disc  is  then  pro- 
portional to  O  B.  If  the  center  of  gravity  is  now  fixed  and  a  force 
proportional  to  OB  acts  on  B  as  shown,  there  will  be  a  twisting 
moment  on  the  disc  proportional  to  OB  X  CD,  Fig.  6.  This 
twisting  moment  is  resisted  by  the  torsional  elasticity  of  the  shaft, 
so  that  there  is  a  tendency  to  produce  torsional^  vibration. 

Since  all  of  the  displacements  are  very  small  the  shaft  may  be 
considered  as  perfectly  straight  in  computing  the  torsional  elas- 
ticity. Now,  both  O  B  and  CD  are  very  small  quantities  as  com- 
pared with  the  torsional  elasticity,  so  that  the  angle  through 
which  the  shaft  is  rotated  by  the  twisting  moment  OB  x  CD  is 
of  the  second  order  of  small  quantities  and  may  be  safely  neglect- 
ed. That  is  to  say,  a  displacement  of  the  disc  has  no  appreciable 
effect  in  causing  additional  rotation  of  the  disc  about  its  center  of 
gravity  beyond  that  due  to  the  constant  angular  velocity. 

There  is  another  possible  displacement  from  the  position  of  Fig. 
5,  consisting  of  a  finite  rotation  of  the  disc  so  as  to  cause  torsion 
in  the  shaft,  as  shown  in  Fig.  7.  If  the  angular  velocity  o>  did 
not  exist,  a  torsional  displacement  would  cause  a  harmonic  tor- 
sional oscillation  of  the  disc  about  its  center  of  grovity.  The 


FIG.  6. 
Transverse 
Vibration. 


FIG.  7. 
Torsional 
Vibration. 


FIG.  8. 
Effect  of 
Gravity. 


FIG.  9 

Most  General 
Vibration. 


twisting  moment  due  to  torsion  of  the  shaft  is  a  pure  couple,  so 
that  it  causes  rotation  about  the  center  of  gravity,  even  though 
the  center  of  bore  is  at  a  different  point.  The  period  of  the  tor- 
sional oscillation  will  be  <f>  =  \f-*  using  the  notation  already 

given.  If  now  the  disc  be  given  a  constant  angular  velocity  w  in 
addition  to  the  torsional  vibration,  the  latter  will  persist  just  as 
before,  so  far  as  rotation  of  the  disc  about  its  center  of  gravity  is 
concerned. 

The  force  due  to  the  net  deflection,  proportional  to  OB,  Fig.  7, 
will  here  also  give  a  twisting  moment  about  the  center  of  gravity 
proportional  to  OB  x  CD.  This  is  of  the  second  order  of  small 
quantities  and  will  be  neglected  as  before. 


-43- 

The  line  B  C,  Fig.  7,  will  therefore  rotate  with  a  constant  an- 
gular velocity  o>,  and  also  oscillate  back  and  forth  so  as  to  make 
a  variable  angle  a  with  the  normal  position  B'  C' .  Since  a  varies 
harmonically  with  period  <£,  we  will  have  a  =  ct  cos  <f>  t.  The  in- 
troduction of  the  constant  angular  velocity  does  not  affect  the  har- 
monic oscillation  as  already  stated. 

Next  let  us  consider  the  effect  of  such  a  torsional  oscillation  on 
the  displacement  of  the  center  of  gravity.  Owing  to  the  previ- 
ously considered  displacements  without  rotation,  the  points^  and 
C  were  displaced  equally  beyond  the  displacement  necessary  to 
balance  the  centrfugal  force,  SQ  that  the  force  acting  on  the  center 
of  gravity  could  be  taken  as  proportional  to  its  own  displacement. 
Now  we  must  consider  that  the  force  acting  on  the  center  of  grav- 
ity is  proportional  to  the  displacement  of  B  beyond  the  position 
necessary  to  balance  the  centrifugal  force.  This  is  not  the  dis- 
placement of  the  center  of  gravity  owing  to  the  variable  angle  a. 

The  position  of  C  without  the  vibration  we  are  now  considering 
would  be  at  a  distance  R  from  O,  Fig.  7,  near  C ',  and  owing  to 
the  addition  of  the  vibration  will  be  supposed  to  be  C  at  any  in- 
stant. The  vibration  we  are  considering  is  the  absolute  vibration 
of  the  center  of  gravity  in  space  in  addition  to  the  constant  rota- 
tion. 

Suppose  we  put  OC'  =  £  =  fi-\-v  and  C  C'  =  rj  and  resolve 
forces  and  accelerations  along  and  perpendicular  to  the  moving 
axis  O  C' .  v  and  rj  will  then  give  the  departures  from  the  equi- 
librium position  due  to  the  vibration  we  are  considering. 

The  force  perpendicular  to  the  axis  is  —  AT  X  B  B'  or  —  K(-rj  + 
flsina),  since  we  have  CC'  =  t]  =  BB'  — a  sin  a.  Since  a  will 
always  be  small  we  take  a  instead  of  sin  a.  We  have  already  re- 
marked that  a  =  rt  cos  <f>  t,  so  that  the  force  perpendicular  to  the 
moving  axis  is  —  K '(rj  -f-  ac^  cos  <f>  t}. 

The  total  force  in  the  direction  of  the  moving  axis  is 

-  K  X  O  C  or  —  K (£  +  a  cos  a). 

Now  a  portion  of  this,  —  K  (R  +  «),  is  required  to  maintain  equi- 
librium against  the  centrifugal  force,  so  that  the  remaining  force, 
—  K ' (y  —  a  -f-  a  cos  a),  is  the  force  in  the  direction  of  the  moving 
axis  tending  to  cause  the  vibration  we  are  considering.  Since  a 
is  small  we  may  for  a  second  approximation  put  cos  a  —  i,  so  that 
the  force  becomes  —  Kv.  The  accelerations  along  and  perpendic- 
ular to  a  moving  axis  are  respectively  (Routh's  Rigid  Dynamics, 
Vol.  I,  page  176), 

d2£  2       i    d  , 


—  44- 
^9_,.«+L  _£«•„) 

Since  w  is  constant  we  may  reduce  the  final  terms.     Also  substi- 
tuting in  terms  of  v,  we  have  as  the  accelerations, 


_-_. 

Now  the  term  —  R  w2  gives  the  acceleration  due  to  the  rotation  in 
the  equilibrium  position  so  that  the  acceleration  due  to  the  added 
vibration  alone  is  found  by  omitting  this  term.  We  have  already 
omitted  the  corresponding  centrifugal  force,  —  K  '(R  +  a). 

Bquating  the  forces  previously  found  along  and  perpendicular 
to  the  moving  axis  to  the  forces  required  to  produce  the  above  ac- 
celerations we  have 


,2  ,  .   .-.       d2  77  o   i          d  v 

—  ^2(77  -fa^COS^O  =          '  —  7?a>2-f  20)-—. 
at  at 

This  is  a  pair  of  simultaneous  linear  differential  equations  whose 
solution  is  complicated.  The  integration  will  be  omitted  and  the 
integrals  only  given.  These  will  be  found  correct  by  substitution 
in  the  equations. 

v  =  2  B  to  <£  sin  <£  / 

y  =  _  B  O2  -  co2  -  <£2)  cos  <f>  t 
where 


Since  v  and  rj  are  displacements  along  and  perpendicular  to  the 
rotating  axis  O  C'  ,  and  since  the  values  above  evidently  repre- 
sent a  harmonic  motion  in  an  ellipse  with  axes  in  these  directions, 
it  follows  that  any  accidental  torsional  displacement  will  cause  a 
vibration  of  the  center  of  gravity  with  period  <£  in  an  ellipse  whose 
axes  rotate  with  an  angular  velocity  w,  one  of  them  always  coin- 
ciding with  the  radius  vector  to  Ihe  equilibrium  position  of  the 
center  of  gravity.  This  is  in  addition  to  the  simultaneous  tor- 
sional oscillation  of  the  disc  and  shaft.  This  oscillation  and  con- 
sequently the  vibration  of  the  center  of  gravity  induced  by  it  will 
of  course  rapidly  die  out  owing  to  molecular  friction,  so  that  the 
equilibrium  position  of  Fig.  5  will  be  again  reached. 


-45- 

We  have  always  supposed  a  vertical  shaft  so  that  gravity 
need  not  be  considered.  Let  us  now  make  the  shaft  hori- 
zontal and  take  gravity  into  account.  First  taking  all  the 
forces  as  acting  on  the  center  of  gravity,  we  will  have  a  down- 
ward pull  M  and  an  equal  upward  pull  due  to  a  down- 
ward deflection  of  the  shaft,  the  displacement  of  the  disc 

M 
being  ^>       The  point  C  will  therefore  rotate  in  a  circle  of  radius 

R  whose  center  is   O  Fig.  8,  a  distance  —    below  the  axis  of  ro- 

JK 

tation  O'. 

Next  let  us  consider  if  rotation  occurs  about  the  center  of  grav- 
ity due  to  the  action  of  the  weight  as  shown  in  Fig  8.  If  the 
center  of  gravity  Cbe  fixed  there  will  be  a  force  proportional  to 
the  net  deflection  O  B  acting  at  B,  tending  to  rotate  the  disc  and 
resisted  by  the  torsional  elasticity  of  the  shaft.  The  twisting 
moment  will  be  proportional  to  O  B  X  CD,  Fig.  8,  and  will 
therefore  be  a  small  quantity  of  the  second  order,  so  that  the 
torsional  vibration  induced  will  be  negligible  just  as  in  the  pre- 
vious cases. 

We  have  now  analysed  the  matter  completely  and  taken  ac- 
count of  everything  which  can  occur.  The  methods  used  have 
been  chosen  so  as  to  show  most  clearly  the  actual  nature  of  the 
various  actions,  and  have  perhaps  been  rather  inelegant  mathe- 
matically. We  will  therefore  go  through  with  the  matter  again 
from  a  strictly  mathematical  standpoint  for  verification  of  our  re- 
sults. This  treatment  is  distinctly  obscure  so  far  as  the  nature 
of  the  action  is  concerned  however. 

Fig.  9  shows  the  most  general  position  of  the  various  points. 
Since  we  have  a  rotation  with  angular  velocity  w,  induced  by  an 
external  source,  we  take  as  a  reference  line  an  axis  rotating  with 
this  velocity  and  therefore  making  an  angle  <o  t  with  the  X  axis 
at  any  instant.  The  most  general  displacement  of  the  shaft 
center  B  will  be  along  and  perpendicular  to  the  rotating  axis. 
We  will  take  as  the  amount  of  the  net  deflection  at  any  instant 
p  —  O  B,  and  suppose  that  it  makes  an  angle  8  with  the  reference 
axis.  Such  a  displacement  would  leave  B  C  parallel  to  the  refer- 
ence axis.  We  will  also  suppose  a  rotation  of  the  disc  and  the 
line  B  C  and  a  twist  of  the  shaft  through  an  angle  a.  We  will  as 
is  usual  first  find  the  motion  of  the  center  of  gravity  by  taking  all 
of  the  forces  as  acting  on  it.  We  will  resolve  forces  along  the  X 
and  Faxes.  At  first  sight  it  would  seem  simpler  to  resolve  forces 


-46- 

along  and  perpendicular  to  the  moving  axis.     However,    the  re- 
sulting equations  turn  out  to  be  much  more  difficult  to  integrate. 
The  force  equations  are 

M  d2  x  „  f    *       *\ 

-  —  -  =  —  K  p  cos  (o>  t  —  8) 

g  dt 


Evidently 

p  cos  (co  /  —  8)  =  x  +  a  cos  (w  t  -f  a) 
p  sin  (w  /  —  8)  =  jj/  +  a  sin  (w  /f  +  a)  . 

Hence,  our  force  equations  may  be  written 

^.=  -  ^  {  x  +  a  cos  (o,  /  +  a)    j 

»2  /*  *\ 

^^  =  —  ^2  J  j/  -f  a  sin  (w  /  +  a)    I  —  g. 

•  In  order  to  integrate  these  we  must  find  an  expression  for  a. 
This  we  do  by  forming  the  equation  of  rotation  about  the  center 
of  gravity,  assuming  that  it  is  fixed  and  that  all  of  the  forces  act 
to  cause  rotation  about  it'. 

The  force  moment  equation  then  is 


We  have 


dt*  dt*' 

Now  A'  si  n  (a  -f  8)  is  of  the  same  order  of  magnitude  as  L  a. 
p  and  a  are  both  small  quantities  so  that  their  product  is  of  the 
second  order  of  small  quantities,  and  therefore  negligible.  Hence 
the  term  Kp  a  sin  (a  -f-  8)  is  negligble  as  compared  with  L  a. 
This  amounts  to  saying  that  the  twisting  moment  A"  X  OB  X 
CD,  Fig.  9  can  never  cause  an  appreciable  torsional  vibration  of 
the  shaft. 

Hence  we  have 


This  is  a  linear  equation  whose  solution  is  well  known  to  be 
a  =  cv  cos  <j>  t. 

There  is  in  general  a  second  arbitrary  constant  which  we  may 
take  zero  without  loss  of  generality,  beginning  our  count  of  time 
from  the  instant  when  the  vibration  is  at  one  extreme. 


—  47- 

It  is  to  be  remarked  that  a  variable  twisting  moment  would 
really  be  required  to  drive  the  shaft  at  the  constant  angular 
velocity  w,  owing  to  the  variation  of  a  and  of  the  shaft  displace- 
ment perpendicular  to  the  reference  line.  Both  of  these  should 
be  balanced  by  a  variable  external  twisting  force  in  order  that 
there  should  be  no  angular  Deceleration  at  the  extremity  of  the 
shaft.  However,  there  are  usually  rotating  bodies  with  consider- 
able mass  geared  to  the  shaft,  and  the  inertia  of  these  serves  to 
prevent  any  appreciable  angular  acceleration. 

Substituting  our  value  of  a  in  the  force  equations  above,  and 
taking  a  as  small  so  that  we  can  put  cos  a=r  i  sin  a  =  a,  we  have 

—  5  -f  ^  x  —  —  if/2  a  cos  w  t  H-  V2  #  *\  cos  <£  t  sin  w  / 

d2  v 

—  5  +  "fy  —  —  ^2  #  sin  <o  /  —  i/r2  a-c^  cos  <f>  t  cos  o>  t  —  g. 

These  are  linear  differential  equations  which  are  integrated  by 
well  known  methods  which  we  need  not  consider  here.  The  form 
to  which  we  reduce  the  integrals  has  been  arranged  to  exhibit 
the  various  individual  vibrations  already  discussed. 

The  integrals  are 
x  =  R  cos  o>  /  -f  c3  cos  ft  cos  O  /  +  ft  )  +  <:3  sin  ft  sin  O  t  +  ft) 


-f-  B  \  (  i/f2  —  to2  —  <£2)  sin  co  t  cos  <f>  t  +  2  co  <£  cos  co  /  sin  ^>  /   !• 
<o  t  +  ^2  sin  ft  cos  (i/'  /  +  ft)  +  ^3  cos  ft  sin  ($  t  -\-  ft) 

—  co2  —  <£2)  cos  <o  /  cos  <£  / — 2  co  c/>  sin  co  t  sin  </)/[•  — 

3       K 


the  arbitrary  constants  of  the  initial  displacements.  These  are 
evidently  the  projections  on  the  J^and  Y  axes  of  the  following  : 
A  rotation  in  a  circle  of  radius  R  with  a  constant  angular  velocity 
o>,  an  elliptica1  vibration  with  axes  fixed  in  space  and  angular 
velocity  ^,  an  elliptical  vibration  with  axes  along  and  perpendic- 
ular to  the  moving  reference  axis  and  angular  velocity  <£,  and  a 
displacement  vertically  downward.  These  are  the  motions  already 
found  otherwise  the  coefficients  also  agreeing.  Hence,  we  con- 
clude finally  that  the  most  general  possible  motion  will  consist  of 
vibrations  about  the  position  of  Fig.  5,  which  will  die  out  ow- 
ing to  molecular  friction  so  that  the  actual  position  of  Fig.  5  will 
always  be  reached. 


CHAPTER  VI. 

EXPERIMENTS  WITH  A  GAS  TURBINE. 

This  portion  of  the  investigation  of  the  gas  turbine,  really  the 
most  important  of  all,  has  unfortunately  turned  cfUt  in  a  very  un- 
satisfactory manner.  So  far  as  principles  are  concerned  all  of  the 
statements  previously  made  were  found  correct,  but  no  actual  re- 
sults were  obtained.  After  many  failures  a  gas  turbine  was 
actually  operated,  but  the  net  output  was  negative.  That  is  to 
say,  the  turbine  wheel  did  not  even  yield  enough  power  to  com- 
press the  air  required.  This  result  is  to  be  attributed  solely  to 
the  crudness  of  the  apparatus  used.  The  apparatus  was  arranged 
very  much  like  the  diagram  of  Fig.  2,  except  that  the  air  was 
supplied  by  an  independent  source  and  the  whole  output  of  the 
wheel  measured  by  a  brake. 

Considerable  difficulty  was  found  in  igniting  the  air  and  oil  and 
starting  the  continuous  combustion  under  pressure.  A  successful 
method  of  doing  this  was  finally  found  however.  Next  trouble 
occured  in  securing  smokeless  combustion,  which  was  also  over- 
come. After  this  the  combustion  gave  no  further  trouble,  and 
gases  at  a  white  hot  temperature  issued  from  the  nozzle  with  per- 
fect regularity.  However,  the  nozzles  first  used  were  soon  de- 
stoyed.  After  many  experiments  a  nozzle  was  built  which  lasted 
a  reasonable  time.  Permanent  success  was  not  attained,  however. 

The  turbine  wheel  used  was  that  from  a  very  small  DeL,aval 
steam  turbine  and  was  about  5  inches  mean  diameter.  The  best 
speed  was  found  to  be  about  19,000  revolutions  per  minute,  and  at 
this  speed  about  3  horsepower  was  developed.  About  4  horse- 
power was  theoretically  required  to  compress  the  air. 

No  experimental  work  was  done  in  the  way  of  varying  the 
pressure,  the  shape  of  the  nozzle,  and  the  shape  of  wheel  buckets. 
There  is  of  course  an  almost  infinite  field  for  variation  in  these 
particulars  and  the  writer  has  no  doubt  that  experiments  in  these 
directions  will  ultimately  be  crowned  with  success. 


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